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THE  UNIVERSITY  OF  CHICAGO 
SCIENCE  SERIES 


Editorial  Committee 

ELIAKIM  HASTINGS  MOORE,  Chairman 

JOHN  MERLE  COULTER 
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The  University  of  Chicago  Science  Series, 
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ously  have  appeared  only  in  scattered  articles, 
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FINITE  COLLINEATION  GROUPS 


THE  UNIVERSITY  OF  CHICAGO  PRESS 
CHICAGO.  ILLINOIS 


Bgenta 

THE  BAKER  &  TAYLOR  COMPANY 

NEW  YORK 

THE  CUNNINGHAM,  CURTISS  &  WELCH  COMPANY 

LOS  ANGELES 


THE  CAMBRIDGE  UNIVERSITY  PRESS 

LONDON  AND  EDINBURGH 

THE  MARUZEN-KABUSHIKI-KAISHA 
THE  MISSION  BOOK  COMPANY 

SHANGHAI 


FINITE  COLLINEATION 
GROUPS  ih.j{. 

WITH  AN  INTRODUCTION  TO  THE  THEORY 

OF  GROUPS  OF  OPERATORS  AND 

SUBSTITUTION  GROUPS 


By 

H.  F.  BLICHFELDT 

Professor  of  Mathematics  in  Leland  Stanford  Junior  University 


THE  UNIVERSITY  OF  CHICAGO  PRESS 
CHICAGO,  ILLINOIS 


^ 


COPYRIGHT  1917  BY 
THE  UNIVERSITY  OF  CHICAGO 


All  Rights  Reserved 


Published  April  1917 


Composed  and  Printed  By 

The  University  of  Chicago  Press 

Chicago.  Illinois.  U.S.A. 


PREFACE 

The  theory  of  finite  collineation  groups  (or  linear 
groups)  as  developed  at  present  is  to  be  found  mainly  in 
scattered  articles  in  mathematical  journals,  in  addition 
to  a  few  texts  on  group  theory.  The  author  has  en 
deavored  in  the  present  volume  to  give  an  outline  of  the 
different  principles  contained  in  these  publications,  and 
has  at  the  same  time  made  an  effort  to  depend  upon  a 
minimum  of  abstract  group  theoiy.  In  this  and  many 
other  respects  the  present  volume  differs  from  Part  II 
of  the  last  book  cited  in  §  24 ;  in  particular,  the  present 
volume  contains  more  of  the  theory  of  linear  groups, 
though  the  student  is  referred  to  that  Part  II  for  lists  of 
the  invariants  of  the  binary  and  ternary  groups. 

No  previous  knowledge  of  the  technique  of  group 
theory  is  required  for  the  reading  of  the  opening  chapter, 
which  develops  the  fundamental  properties  of  linear 
transformations  and  linear  groups.  For  the  greater 
convenience  of  the  student,  an  introduction  to  the  theory 
of  groups  of  operators  and  substitution  groups  is  given 
in  the  second  chapter;  moreover,  certain  theorems  and 
definitions  from  the  more  advanced  parts  of  algebra 
needed  throughout  the  book  are  stated  in  explicit  form 
in  an  Appendix. 

The  groups  in  two  variables  are  determined  in  chap, 
iii  by  a  modified  form  of  a  process  due  to  Klein,  which 
depends  largely  upon  geometrical  intuition.  Theorems 
which  serve  as  a  means  for  determining  the  relatively 
difficult  linear  groups  in  more  than  two  variables  are 
presented  in  chap,  iv,  and  are,  in  fact,  made  use  of  in 

781V4O7 


viii  PREFACE 

the  chapters  on  ternary  and  quaternary  groups  (chaps,  v 
and  vii). 

Concerning  the  theory  of  group  characteristics 
(chap,  vi),  an  attempt  has  been  made  to  exhibit  this  sub 
ject  in  a  very  simple  form,  by  means  of  explicit  definitions 
and  easy  proofs  which  eliminate  complicated  sigma- 
constructions.  The  student  is  here  (as  elsewhere)  urged 
to  work  with  the  matrix  form  of  a  linear  transformation 
and  intransitive  group  (§§  2,  85). 

Where  purely  geometrical  methods  are  not  available 
or  are  less  convenient  than  analytical  methods,  col- 
lineation  groups  are  most  easily  studied  through  the 
medium  of  " linear  groups"  (cf.  §§  10,  51);  hence  the 
almost  exclusive  discussion  of  the  latter  category  in 
the  text. 

The  author  owes  his  thanks  to  his  colleagues,  Profes 
sors  R.  E.  Allardice  and  W.  A.  Manning,  for  the  care 
with  which  they  have  read  the  manuscript  and  for  many 
helpful  criticisms.  He  is  also  deeply  indebted  to  Pro 
fessors  E.  H.  Moore  and  L.  E.  Dickson  of  the  University 
of  Chicago  for  their  generous  encouragement  and  valuable 
suggestions. 

H.  F.  BLICHFELDT 

LELAND  STANFORD  JUNIOR  UNIVERSITY 


TABLE  OF  CONTENTS 

CHAPTER  PAGE 

I.  ELEMENTARY  PROPERTIES  OF  LINEAR  GROUPS  ....         1 

§§  1-6,  Linear  transformations:  definitions,  funda 
mental  properties.  §§  7-14,  Groups  of  linear  trans 
formations:  classification.  §  13,  Change  of  variables. 
§  14,  Transitive  and  intransitive  groups.  §§  15-20, 
Hermitian  invariant  and  reducibility  of  linear  groups. 
§  19,  Unitary  transformations.  §  20,  Reducible  and 
irreducible  groups.  §§  21-22,  Canonical  form  of  a 
linear  transformation  and  of  abelian  groups.  §  23, 
Characteristic  and  chacteristic  equation. 

II.  GROUPS  OF  OPERATORS  AND  SUBSTITUTION  GROUPS      29 

§  24,  Introduction.  §§  25-39,  Groups  of  operators. 
§§  25-26,  Operators.  §§  27-28,  Finite  groups;  gen 
erators;  subgroups.  §§  29-31,  Conjugate  sets  and 
subgroups;  invariant  subgroups;  simple  groups. 
§§32-33,  Isomorphism;  factor  groups.  §34,  Abe 
lian  groups.  §§  35-39,  Groups  whose  orders  are 
powers  of  a  prime  number.  §§  36-38,  Sy low's 
theorem.  §§  40-45,  Substitution  groups.  §§  40-42, 
Definitions;  notation;  even  and  odd  substitutions. 
§  43,  Substitution  groups;  symmetric  and  alternating 
groups.  §§  44-45,  Transitive  and  intransitive  groups; 
theorem.  §§  46-47,  On  the  representation  of  a  group 
of  operators  as  a  substitution  group;  regular  group. 
§§  48-50,  On  simple  groups.  §§  49-50,  Theorems  on 
the  alternating  group. 

III.  THE  LINEAR  GROUPS  IN  Two  VARIABLES 63 

§  51,  General  remarks  on  linear  groups  and  collineation 
groups;  equivalence.  §§  52-55,  Determination  of  the 
linear  groups  in  two  variables.  §§  56-58,  The  groups 
of  the  regular  polyhedra.  §  59,  Jordan's  process: 
the  diophantine  equation, 
ix 


X  TABLE  OF  CONTENTS 

CHAPTER  PAGE 

IV.  ADVANCED  THEORY  OF  LINEAR  GROUPS 76 

§  60,  Primitive  and  imprimitive  groups.  §§  61-62,  On 
the  form  of  Sylow  subgroups;  theorems.  §§  63-74, 
On  the  order  of  primitive  groups.  §§  63-64,  Superior 
limit  to  the  magnitude  of  the  prime  factors.  §§  65- 
68,  Superior  limit  to  the  factors  which  are  powers  of  a 
prime  number;  theorem  on  two  commutative  trans 
formations.  §§  69-73,  Superior  limit  to  the  order  of 
an  abelian  subgroup.  §  74,  Superior  limit  to  the 
order  of  a  primitive  group  in  n  variables;  historical 
note. 


V.  THE  LINEAR  GROUPS  IN  THREE  VARIABLES 104 

§  75,  Introduction.  §§  76-77,  The  intransitive  and  im 
primitive  groups.  §§  78-79,  Primitive  groups  hav 
ing  invariant  intransitive  or  imprimitive  subgroups. 
§§  80-82,  The  primitive  simple  groups.  §  83,  Primi 
tive  groups  having  invariant  primitive  subgroups; 
bibliography. 


VI.  THE  THEORY  OF  GROUP  CHARACTERISTICS 116 

§§84-88,  Introduction;  definitions;  the  sum  of 
matrices;  invariants.  §§  89-91,  On  the  character 
istics  of  transitive  groups.  §§  92-94,  On  the  char 
acteristics  of  isomorphic  groups;  composition  of  two 
groups.  §§  95-99,  On  the  totality  of  non-equivalent 
isomorphic  groups;  the  regular  group.  §  100,  An 
application:  no  group  of  order  p"qb  can  be  simple. 


VII.  THE  LINEAR  GROUPS  IN  FOUR  VARIABLES 139 

§101  Introduction.  §§  102-17,  The  primitive  simple 
groups.  §§  118-19,  Groups  which  contain  primi 
tive  simple  groups  as  invariant  subgroups.  §§  120- 
25,  Non-primitive  groups  and  primitive  groups 
which  contain  invariant  non-primitive  subgroups. 


TABLE  OF  CONTENTS  xi 

CHAPTEK  PAGE 

VIII.  ON  THE  HISTORY  AND  APPLICATIONS  OF  LINEAR 

GROUPS 174 

§  126,  The  history  of  linear  groups.  §  127,  Klein's  ex 
tension  of  the  Galois  theory  of  equations.  §  128,  The 
connection  between  linear  differential  equations  hav 
ing  algebraic  solutions  and  linear  groups. 


APPENDIX 183 

§§  129-31,  On  congruences  and  indeterminate  equa 
tions.  §  132,  On  the  value  of  a  certain  determinant. 
§  133,  On  roots  of  unity.  §  134,  On  algebraic  inte 
gers. 


INDEXES  .  191 


CHAPTER  I 

ELEMENTARY  PROPERTIES  OF  LINEAR  GROUPS 
LINEAR  TRANSFORMATIONS:    FUNDAMENTAL 

PROPERTIES,    §§  1-6 

1.  Examples  of  linear  transformations.—  Operator. 
Let  the  axes  of  co-ordinates  (rectangular)  in  the  plane 
be  rotated  through  an  angle  0,  the  origin  remaining  fixed. 
If  x,  y  are  the  co-ordinates  of  any  given  point  with  refer 
ence  to  the  axes  in  their  original  position,  and  x',  y'  the 
co-ordinates  of  the  same  point  with  reference  to  the  axes 
in  their  new  position,  then,  as  is  proved  in  analytic 
geometry, 

,  ,  x  =  x'  cos  0  —  y'  sin  0  , 

y  =  x'  sin0-fy  cos0. 

Hence,  if  a  given  curve  has  for  its  equation 

(2)  /(*,») -0 

with  reference  to  the  old  axes,  then  its  equation  with 
reference  to  the  new  axes  will  be 

(3)  f(x'  cos  0-yf  sin  0,  x'  sin  B+y'  cos  0)  =  0 . 

We  shall  say  that  (3)  is  obtained  from  (2)  by  the 
linear  transformation  (1).  In  conformity  with  the  general 
group  terminology  we  call  the  transformation  (1)  an 
operator,  which,  operating  upon  (2),  produces  (3). 

To  take  a  second  example:  any  literal  substitution  (cf.  chap,  ii, 
B)  can  be  exhibited  as  a  linear  transformation.  For  instance,  the 
substitution  (xix&»)  (x4x5)  can  be  written 


2  FINITE  COLLINEAT1ON  GROUPS 

2.  Formal  definition.     A  linear  transformation  in  n 
variables  is  a  set  of  linear  homogeneous  equations, 


(4)  x2 

.    .    .    +annx 


expressing  the  original  variables  x\,  .  .  .  ,  xn  in  terms  of 
new  variables  x[,  .  .  .  ,  x,[,  under  the  condition  that  the 
equations  can  be  solved  for  the  latter;  that  is,  the  determi 
nant  of  the  coefficients  ast  (s,  t=  1,  2,  .  .  ,  n),  called  the 
determinant  of  the  transformation,  must  not  vanish. 

We  represent  a  linear  transformation  by  a  capital 
letter  (as  A,  S,  .  .  ),  or,  if  we  wish  to  be  specific,  by  the 
matrix  of  the  linear  transformation 

[an  ai2  .      .     ain~\ 
ani  anZ  .      .     ann\ 

which  may  be  abbreviated  to  [ast\.  Thus  the  equation 
A  =  [ast]  implies  that  a  transformation  denoted  by  A  has 
for  coefficients  the  numbers  ast  (s,  t=l,2,  .  .  ,  n).  The 
same  implication  is  also  indicated  by  writing  A:  to  the 
left  of  the  equations  (4)  .  The  equation  A  =  B  is  equiva 
lent  to  the  statement  that  the  matrices  of  A  and  B  are 
identical. 

We  do  not  a  priori  place  any  restrictions  on  the  form  of  the 
numbers  ast-  They  may  be  real  or  imaginary,  rational  or  irrational. 

The  transformation  (4),  which  we  shall  denote  by  A, 
operates  upon  a  given  function  f(x\,  .  .  ,  xn)  by  putting 
in  place  of  x\,  .  .  ,  xn  the  equivalent  expressions  in  the 
right-hand  members  of  (4).  This  is  indicated  symboli- 


PROPERTIES  OF  LINEAR  GROUPS  3 

cally  by  (f)A.  However,  for  reasons  apparent  later,  we 
shall  drop  the  accents  after  the  operation  has  been  per 
formed.  Thus,  if  S  is  the  transformation  (1)  and  /  the 
function  (2),  the  left-hand  member  of  the  equation  (3) 
will  be  written 

(f)S=f(x  cosB-y  sin  0,  x  sin  0+y  cos  0) 

in  the  future. 

Certain  special  forms  of  linear  transformations  occur 
frequently  or  are  introduced  for  convenience: 

(a)  Canonical  form. — If  every  ast  =  0  in  (4)  except 
when  s  =  t,  we  say  that  A  has  the  canonical  form.  In  this 
case  we  write  A  =  (au,  #22,  .  .  ,  a«n).  The  coefficients 
On,  #22,  .  .  ,  ann  are  here  called  the  multipliers  of  A. 

The  transformations  A  2,  E,  B^  B*,  §  7,  all  have  the  canonical 
form. 

(6)  Similarity-transformation. — This  is  a  transforma 
tion  in  canonical  form  all  of  whose  multipliers  are  equal 
(an  =  a22=  .  .  =a/m),  as  in  the  cases  A2,  E,  §  7. 

(c)  The  identity. — This  is  a  similars-transformation 
whose  multipliers  are  unity. 

We  shall  reserve  the  letter  E  for  this  transformation: 
#=(1,1,  .  .  ,1).  Evidently,  the  identity  produces  no 
change;  that  is,  (f)E=f.  For  an  illustration,  put  0  =  0 
in  (1). 

3.  The  product  of  linear  transformations.  If  we 
operate  on  a  given  function/  by  two  linear  transformations 
successively,  say  first  by  A  and  then  by  B,  the  result, 
which  is  written  (f)AB,  is  equivalent  to  that  obtained  by- 
operating  on  /  by  a  single  linear  transformation  C.  For 
instance,  let/  be  a  function  f(x,  y),  A  the  transformation 
(1),  §  1,  and  B=(a,  6),  and  we  find  x  -  ^  y* 

(f)AB=f(xa  cos  B-yb  sin  0,  xa  sin  0+yb  cos  0), 


4  FINITE  COLLINEATION  GROUPS 

which  is  obviously  also  the  result  of  operating  on/(z,  y)  by 
C:  x  =  x'a  cos  6  —y'b  sin  6,  y  =  x'a  sin  0+y'b  cos  0. 

Remark.  —  We  notice  that 

(f)BA  =f(xa  cos  0  —  ya  sin  0,  xb  sin  6-\-yb  cos  0), 
so  that  (f)AB^(f)BA  unless  (a-b)  sin  0  =  0. 

THEOREM  1.  LeZ  A  =  [a8t]  and  B  =  [b8t]  be  two  linear 
transformations  in  n  variables.  Operating  first  by  A  and 
then  upon  the  result  by  B  is  equivalent  to  operating  originally 
by  a  single  linear  transformation  C  =  [cst]  in  n  variables, 
where 


+asnbnt 


We  say  that  C  is  the  product  of  A  and  B,  and  write  symboli 
cally  AB=C* 

The  rule  for  finding  the  product  AB  =  C  may  be  described  in  the 
following  manner  (matrix  multiplication).  We  define  the  product 
of  a  row  (horizontal)  of  A,  say  «2i,  «22,  .  -  ,  azn,  and  a  column 
(vertical)  of  B,  say  613,  623,  -  -  ,  bm  as  follows:  multiply  the  first 
element  of  the  row  by  the  first  element  of  the  column,  the  second 
element  of  the  row  by  the  second  element  of  the  column,  etc.,  and 
add  the  resulting  n  products  (021613+022623+  .  .  +«2n6/t3  in  the 

*  A  value  for  cst  different  from  that  given  above  is  obtained  by 
writing  last  in  the  product  that  transformation  which  operates  first,  as 
is  the  custom  with  functional  operators  (BA(f)),  or  by  regarding  the 
accented  letters  in  (4),  §  2,  as  the  old  variables  and  the  unaccented  as 
the  new.  Thus,  Klein,  Jordan,  Burnside,  and  Weber  obtain 

_,*  =  n 
cst  =  Zk  =  l  «*«***• 

while  Schur  and  Probenius  get  the  value  given  in  (5).  The  author  has 
hitherto  (in  papers  published  in  technical  journals)  accented  the  old  and 
left  unaccented  the  new  variables,  and  has  used  the  term  "linear  substitu 
tion"  for  "linear  transformation." 


PROPERTIES  OF  LINEAR  GROUPS  5 

example).     Then  cst  is  the  product  of  the  sth  row  of  A  and  the  tth 
column  of  B. 

The  value  of  cst  given  above  is  found  by  eliminating  x(,  x'z,   .    .    , 
x'n  from  the  two  sets  of  equations 


A: 

B:  x's  =  bslxfl'+   .    .    .    +bsnxf^  (s  =  l,2,   .    .   ,  n). 


4.  The  commutative  and  associative  laws.  Power 
of  a  linear  transformation.  In  general,  the  commuta 
tive  law  does  not  hold;  that  is,  AB  differs  from  BA  (cf. 
Remark,  §  3).  On  the  other  hand,  the  associative  law 
holds  in  a  product  of  three  or  more  transformations. 
Thus,  let  A,  B,  C  be  any  three  transformations,  and  let 
AB  =  P  and  BC  =  Q.  Then  (AB)C  =  PC  =  AQ  =  A(BC). 
As  a  consequence  the  notation  ABC  is  not  ambiguous,  and 
we  shall  write  A2  for  AA,  A3  for  A2A,  etc.  We  call  Am 
the  rath  power  of  the  transformation  A. 

To  prove  the  associative  law,  we  construct  the  matrices  of 
p  =  [pst]=AB,  Q  =  [qst]=BC  by  Theorem  1,  §3,  and  then  those  of 
PC  and  AQ. 

Linear  transformations  having  the  canonical  form  are 
commutative  and  their  products  are  readily  written  down. 
In  particular,  if 

S=(an,  «22,   .    .   ,  ann),  T=(bn,  622,   .    .   ,  bnn), 

we  have 

'      ST=TS=(aubn,  a22622,   .    .   ,  annbnn), 
and 


5.  The  inverse  of  a  linear  transformation  S  is  such  a 
transformation,  written  S~l,  that  the  product  SS~1  is 
equivalent  to  the  identity  E;  i.e.,  produces  no  final  change. 
In  the  case  of  the  transformation  (1),  §  1,  the  inverse  is 


6  FINITE  COLLINEATION  GROUPS 

plainly  the  rotation  of  the  axes  through  the  angle  —  0: 
x  =  x'  cos  B-\-yf  sin  0,  y  =  —  x'  sin  6-\-yf  cos  0.  It  is  also 
plain  that,  having  transformed  a  function  f(x,  y)  by  means 
of  (1)  into  a  function  F(x',  y'),  we  get  the  original  function 
f(x,  y)  by  substituting  in  F  the  values  of  x',  y'  expressed 
in  terms  of  x,  y.  These  solutions  appear  in  the  form  of  a 
linear  transformation  after  the  accents  have  been  properly 
placed. 

THEOREM  2.     The  inverse  of  a  linear  transformation  S: 
xs  =  asix'+    .    .   +asnXn      (s  =  l,  2,   .    .   ,  w), 

is  obtained  by  solving  this  system  of  linear  equations  for  xl, 
x(,  .  .  ,  x'n  in  terms  of  Xi,  x2,  .  .  ,  xn.  After  the 
accents  have  been  properly  placed,  these  solutions  appear  in 
the  standard  form  of  a  linear  transformation,  denoted  by  S~l. 

More  generally,  we  shall  denote  the  inverse  of  Sm  by 
S~m,  and  we  have  SmS-m  =  S-mSm  =  E. 

We  can  now  prove  that,  given  AB  =  AC,  then  B  =  C. 
For,  from  A~1(AB)=A~l(AC)  follows  (§4): 


Similarly,  if  BA  =  CA,  then  B  =  C. 

6.  Order  of  a  linear  transformation.  Consider  the 
transformation  (1),  §  1,  which  we  shall  indicate  by  S. 
If  6  is  an  aliquot  part  of  2?r,  then  some  power  of  this 
transformation  will  be  equivalent  to  the  identity.  Thus, 
if  0  is  a  right  angle,  S*  =  E.  In  general,  if  S  is  an  arbi 
trarily  chosen  transformation,  the  identity  will  not  be 
found  among  the  powers  of  S.  If  it  should  be,  we  say 
that  S  is  of  finite  order;  and  if  m  is  the  lowest  positive 
integer  for  which  tim  =  E,  we  say  that  S  is  of  order  m. 
In  this  case  no  power  of  S  need  be  taken  higher  than  m. 


PROPERTIES  OF  LINEAR  GROUPS  7 

For,  since  Sm+a  =  SmSa,  and  the  operator  Sm  produces 
no  change,  it  follows  that  Sm+a  =  Sa. 

If  S  is  of  order  m,  the  order  of  Sh  is  m/d,  where  d 
is  the  highest  common  factor  of  ra  and  k  (cf.  §  26,  (6)). 
Thus,  if  m=12,  then  S5,  S7  and  S11  are  of  order  12;  S4 
and  >S8  of  order  3,  and  so  on. 

EXERCISES 

1  .  If  di  and  d2  are  the  determinants  of  the  linear  transformations 
A  i  and  Az  in  two  variables,  then  the  determinant  of  A\AZ  is  d\d2. 

2.  Prove  that  the  product  of  a  linear  transformation  T  and  a 
similarity-transformation  S  =  (a,  a,   .    .   ,  a)  is  obtained  by  merely 
multiplying  every  element  in  the  matrix  of  T  by  a. 

Hence  prove  that  a  similarity-transformation  is  commutative 
with  every  transformation  in  the  same  variables. 

3.  Find  the  general  form  of  a  linear  transformation  in  three 
variables  which  is  commutative  with  S  =  (a,  a,  6). 

4.  Prove  that  the  two  transformations 


are  each  the  inverse  of  the  other. 

5.  If  S  is  of  order  m,  then  Sm~~1  is  the  inverse  of  S. 

6.  A  linear  transformation  is  the  inverse  of  its  own  inverse. 

7.  Are  the  following  transformations  of  finite  order: 


en- 


8.  Prove  that  the  multipliers  of  a  transformation  of  finite  order 
are  roots  of  unity  (cf.  §  133). 


GROUPS    OF    LINEAR    TRANSFORMATIONS  I     CLASSIFICATION, 
CHANGE    OF    VARIABLES,    §§  7-14 

7.  Finite  groups  of  linear  transformations.  Let  there 
be  given  a  set  of  distinct  linear  transformations,  finite  in 
number,  and  let  it  be  known  that  the  product  of  any  two 


8  FINITE  COLLINEATION  GROUPS 

of  the  set  (AB  as  well  as  BA),  whether  alike  or  distinct, 
is  again  a  transformation  of  the  set;  then  this  set  is  called 
a  finite  group  of  linear  transformations.  We  shall  usually 
employ  the  simpler  expression  linear  group.  The  number 
of  distinct  transformations  in  the  set  (including  the  iden 
tity)  is  called  the  order  of  the  group. 

As  an  example  of  such  a  group  take  the  four  trans 
formations  consisting  of  the  rotations  of  the  X  —  ,  Y  — 
axes  through  1,  2,  3,  and  4  right  angles  around  the  origin 
in  their  plane: 

0  -n          r-i     on  r   o   11 

1  oj'  A*=\    o  -ij>   M-i   oj> 

-B  a- 

For  a  second  example,  take  the  eight  transformations 
consisting  of  the  four  rotations  just  given,  in  addition 
to  these  four  accompanied  by  a  reflexion  on  the  new  Y  — 
axis: 

Ai,  A2,  A3,  E; 


-n  a- 

The  former  group  is  a  subgroup  of  the  latter. 

The  set  of  three  transformations  A  i,  A2,  Az  do  not  form  a  group, 
since  the  transformation  A\  (or  AI  A3)  is  not  found  in  the  set. 

Notation.  —  We  shall  reserve  the  letters  G,  H,  K  to 
denote  particular  groups  that  may  come  under  discussion. 
The  equation  G  =  (Si,  Sz,  .  .  ,  Sg)  implies  that  a  given 


PROPERTIES  OF  LINEAR  GROUPS  9 

group  G  consists  of  (or  contains)  the  transformations  S\t 

S2,        .         .        ,     Sg. 

It  is  often  convenient  to  use  the  phrase  "  (a  group)  G"  with  the 
same  meaning  as  the  phrase  "the  transformations  of  (a  group)  G." 
For  instance,  we  may  say  "(something)  is  unaltered  by  G"  instead 
of  "(something)  is  unaltered  by  the  transformations  of  G." 

8.  Elementary  properties  of  linear  groups. — Gen 
erators.  Let  G  represent  a  given  linear  group,  and  S 
any  transformation  contained  in  G.  Then  it  is  easy  to 
show  that  S  is  of  finite  order,  say  ra,  and  that  its  different 
m  powers  (including  its  inverse  and  the  identity)  are  con 
tained  in  G.  For,  the  series  of  transformations 

S,  SS  =  S2,  SS*  =  S*,  .   .   . 

all  belong  to  G  and  can  evidently  not  form  an  unlimited 
number  of  distinct  transformations,  the  group  being 
finite.  Hence,  at  least  two  of  these  powers  are  equivalent, 
say  /S°  =  £0+m,  which  may  be  written  SaE=SaSm.  It  fol 
lows  that  E=Sm  (§5).  The  transformation  S  is  there 
fore  of  finite  order  (§6),  and  /S"1"1  is  its  inverse. 

It  may  happen  that  the  various  powers  of  S  exhaust 
the  transformations  of  G.    We  then  say  that  G  is  generated  \ 
by  S,  and  that  S  is  a  generator  of  G.     If  G  contains  otner   ' 
transformations,  let  T  be  one  such,  and  the  products 
SaTb,  SaTbSc,   ...  all  belong  to  G.    We  may  be  able 
to  get  all  the  transformations  of  G  in  this  manner;  if  so, 
we  say  that  S  and  T  generate  G,  or  that  they  form  a  set  of 
generators  of  G;  and  so  on. 

Consider  for  example  the  group  (6),  §  7.  Here 
A\  =  AZ,  Al  =  A3,  A\  =  E.  Hence  AI  generates  this  group, 
which  is  also  geometrically  evident.  In  the  case  of  (7)  we 
have  additional  relations  A iBi  =  B2,  A\Bi  =  Bz,  A\B\  =  B^ 
so  that  AI  and  BI  form  a  set  of  generators  for  this  group. 


10  FINITE  COLLINEATION  GROUPS 

Having  given  a  linear  group,  a  set  of  generators  may 
usually  be  selected  in  different  ways.  It  should  be  noted 
that  a  linear  transformation  selected  at  random  will  not 
generate  a  finite  group  (§6),  and  that  two  or  more  linear 
transformations  each  of  finite  order,  but  otherwise  taken 
at  random,  will  not  generate  a  finite  group. 

9.  Collineations  and  collineation  groups.  If  x\,  .  .  , 
xn  represent  homogeneous  co-ordinates  in  space  of  n  —  1 
dimensions,  then  the  geometrical  effect  of  a  linear  trans 
formation  is  not  altered  by  multiplying  all  the  elements 
in  its  matrix  by  an  arbitrary  constant.  To  illustrate, 
if  Xi,  £•;,  x3  represent  trilinear  co-ordinates  of  the  plane, 
the  transformations 


A  = 


are   both   equivalent   to   the   projective   transformation 
leaving  fixed  the  straight  lines 


and  transforming  the  point  (1,  —  2,  3)  into  the  point 
(1,  0,  1).  We  say  that  A  and  A'  represent  the  an,me 
collineation;  that  is,  a  collineation  is  specified  by  the 
mutual  ratios  of  the  elements  in  the  corresponding  matrix, 
not  by  the  actual  values  of  these  elements  (to  a  given  non- 
vanishing  element  may  therefore  at  the  outset  be  assigned 
at  will  any  convenient  number  not  zero).  In  practice  it  is 
customary  to  affix  a  factor  of  proportionality  to  either  the 
old  or  the  new  variables  to  distinguish  a  collineation  from 
a  linear  transformation;  the  collineation  represented  by 
the  two  transformations  above  will  thus  be  written 


PROPERTIES  OF  LINEAR  GROUPS  11 

The  following  laws  obtain : 

(a)  If  A  and  A'  are  linear  transformations  represent 
ing  the  same  collineation,  then  there  is  a  similarity- 
transformation  S  such  that  A'  =  AS  =  SA  (or  A'A~l  = 
A~1A'  =  S).  Thus,  in  the  example  above,  £=(2,  2,  2). 
Conversely,  S  being  any  similarity-transformation,  A 
and  AS  represent  the  same  collineation. 

(6)  If  A  and  A'  represent  the  same  collineation  Ci, 
and  similarly  B  and  B'  represent  the  same  collineation 
C2,  then  A  B  and  A  'B'  represent  the  same  collineation 
CiC,. 

For,  let  S  and  T  be  the  two  similarity-transformations  A' A-1 
and  B'B~\  so  that  A'  =  AS,  B'  =  BT,  and  we  get  A'B'  =  ASBT  = 
ABST,  since  S  is  commutative  with  B  (§  6,  Exercise  2).  Hence, 
(:4B)-1(A'JB')  =  (AJB)-1(^jBAS77)  =  (AJB)-1(^^)(571)=/S77,  which  is 
a  similarity-transformation  (§4). 

Hence,  to  find  the  product  of  two  collineations,  C\ 
and  C2,  we  take  the  product  of  any  two  representative 
linear  transformations.  If  therefore  we  have  a  finite 
set  of  collineations  such  that  the  product  of  any  two  of 
the  set  (whether  alike  or  different)  is  a  collineation  of  the 
set,  we  call  this  set  a  collineation  group  whose  order  is  the 
number  of  distinct  collineations  of  the  set. 

10.  The  collineation  group  derived  from  a  given 
linear  group.  Let  G  be  a  linear  group  of  order  g, 
and  A  any  one  of  its  transformations.  Furthermore,  let 
K=(Si,  82,  .  .  ,  Sk)  be  the  group  consisting  of  all  the 
similarity-transformations  contained  in  G  (§  11,  Exercise 
1).  Then  (§9,  (a)) 

ASi,  AS2,  .    .  ,  ASk 

are  distinct  linear  transformations,  all  representing  the 
same  collineation,  say  C".  Moreover,  no  further  trans 
formations  of  G  can  represent  C'. 


12  FINITE  COLLINEATION  GROUPS 

If  now  B  is  a  new  transformation  of  G,  we  get  a  new 
collineation  C"  corresponding  to  the  transformations 


Proceeding  thus,  we  shall  finally  arrange  all  the  g  trans 
formations  in  g/k  classes,  giving  rise  to  a  set  of  g/k 
distinct  collineations.  These  form  a  group  of  order  g/k, 
since  the  product  of  any  two  of  them  belongs  to  the  set 
(cf.  §  9,  (6)).  We  formulate  this  result  as  follows: 

THEOREM  3.  To  a  given  linear  group  G  of  order  g 
corresponds  a  collineation  group  Gf  of  order  g/k,  where  k 
is  the  order  of  the  group  of  similarity-transformations  K 
contained  in  G.  To  a  given  collineation  correspond  k 
linear  transformations  of  G,  obtained  from  one  of  them  by 
multiplying  it  in  turn  by  each  of  the  transformations  of  K. 

Example.  —  Let  G  be  the  group  (6),  §  7,  of  order  4. 
Here  K=(A2,  E),  and  we  have  two  collineations  in  G', 
represented  respectively  by  (A2,  E)  and  (Ai,  A3): 

E':  px=    x',    py  =  yf  ; 
A{:  px=-y',    py  =  x'. 

If  a  linear  group  G  of  order  g  contains  no  similarity- 
transformation  other  than  the  identity,  then  G  will  itself 
represent  the  corresponding  collineation  group.  In  other 
cases  it  may,  or  may  not,  contain  a  subgroup  of  linear 
transformations  of  order  g/k  which  represents  the  col 
lineation  group  corresponding  to  G.  For  instance,  there 
is  no  linear  group  of  order  2  contained  in  (6)  whose 
collineation  group  is  that  one  given  in  the  example  above. 
On  the  other  harud,  the  group  E=(l,  1),  A2=(—  1,  —1), 
£2=(1,  —1),  #4=(  —  1,  1)  contains  a  subgroup  which 
may  be  taken  as  its  collineation  group,  namely  E,  J52. 


PROPERTIES  OF  LINEAR  GROUPS  13 

11.  Linear  fractional  groups.  A  collineation  in  n  variables 
Xi,  .  .  ,  xn  may  be  represented  as  a  linear  fractional  transformation 
in  the  n-\  ratios  yi  =  xi/xn,  y2  =  x2/xn,.  .  ,yn-i  =  xn-i/xn.  Assum 
ing  for  simplicity  n  =  3,  the  collineation  corresponding  to  the  linear 
transformation 


as  a  transformation  in  the  variable  yi,  y2,  takes  the  form 
2/1 


auy( 


~  any[+  a32y'2+a33' 
The  transformations  of  K  (cf  .  §  10)  will  all  become  the  identity 


and  the  k  transformations  representing  a  single  collineation  will 
give  rise  to  a  single  linear  fractional  transformation.  We  thus 
obtain  a  linear  fractional  group  of  order  g/k  which  is  simply  iso- 
morphic  (cf  .  §  32)  with  the  collineation  group  G'  and  may  be 
regarded  as  its  equivalent. 

EXERCISES 

1.  Prove  that,  in  a  linear  group  G,  those  transformations  which 
have  the  canonical  form  make  up  a  group  by  themselves.     More 
particularly,  the  similarity-transformations  make  up  a  group  K 
which  is  invariant  under  G  (cf.  §  31). 

2.  If   G  and   G'   are   a  linear  group   and   its   corresponding 
collineation  group,  then  to  the  identity  of  G'  correspond  all  the 
similarity-transformations  of  G. 

3.  From  the  formulas  for  the  product  of  two  linear  transforma 
tions  (§3)  and  the  product  of  two  determinants,  prove  that  the 
product  of  the  determinants  of  two  transformations  A  and  B  is  equal 
to  the  determinant  of  the  transformation  AB  (cf.  Exercise  1,  §  6). 

4.  The  determinant  of  the  linear  transformation  A™  is  the  rath 
power  of  the  determinant  of  A  (cf.  Exercise  3).     Hence  prove  that 
the  determinant  of  a  transformation  belonging  to  a  linear  group  is 
a  root  of  unity  (cf.  §  133).     In  particular,  the  determinant  of  a 
transformation  of  the  third  order  is  1,  w  or  w2,  where  w3  =  l. 

5.  Construct  the  collineation  group  corresponding  to  the  group 
(7),  §  7.     Show  that  there  is  no  subgroup  of  (7)  of  order  4  which 
may  represent  this  collineation  group. 


14  FINITE  COLLINEATION  GROUPS 

12.  Groups  of  linear  transformations  of  determinant 
unity.  The  problem,  having  given  a  collineation  group 
G'  of  order  g'  in  n  variables,  to  construct  a  corresponding 
linear  group  G,  admits  of  an  unlimited  number  of  solutions 
(cf.  §  9,  (a)).  We  shall  limit  the  problem  by  requiring 
the  determinants  of  the  linear  transformations  of  G  to  be 
unity,  and  we  shall  show  that  under  this  condition  the 
order  of  G  is  not  greater  than  ngr. 

Let  A'  be  a  linear  transformation  representing  one 
of  the  collineations  of  G',  and  let  its  determinant  be  d. 
We  then  multiply  it  in  turn  by  each  of  the  n  similarity- 
transformations  Si,  Sz,  .  .  ,  Sn,  where 

Sj=(rj>  TJ,   .    .  ,  TJ), 

r\,  7*2,  .  .  ,  rn  being  the  n  different  roots  of  the  equation 
rnd=l.  The  n  transformations  so  obtained  all  have  a 
determinant  =  1 ;  moreover,  no  linear  transformation 
outside  these  n  will  be  of  determinant  unity  and  will 
represent  the  same  collineation  as  A'  (§  9,  (a)). 

Taking  each  of  the  gr  collineations  in  turn,  we  shall 
have  constructed  a  table  like  that  in  §  10,  containing 
ng'  linear  transformations  in  all.  If  A',  Br,  C'  are  three 
collineations  in  G'  such  that  A'B'  =  C',  then  will  the  prod 
uct  of  any  transformation  of  our  table  from  the  line 
corresponding  to  A'  and  any  transformation  from  the 
line  corresponding  to  Bf,  necessarily  be  a  transformation 
from  the  line  corresponding  to  C",  since  the  determinant 
of  this  product  is  unity  (Exercise  3,  §  11).  In  other 
words,  the  ng'  linear  transformations  form  a  linear  group 
G,  whose  collineation  group  is  G'. 

Example.— Let  £'  =  (#,  A),  where  A  =  (l,  -1).  From 
E  we  obtain  the  two  transformations  E,  Ei  =  (—l,  —  1); 
and  from  A  the  two  AI  =  (  —  i,  i),  A2=(i,  —  i),  where 
i=V-\.  Therefore  G=(E,  Ely  Ai,  A2). 


PROPERTIES  OF  LINEAR  GROUPS  15 

It  may  happen  that  a  group  Gi  of  transformations  of 
determinant  unity  exists  whose  order  is  lower  than  ng' 
and  whose  collineation  group  is  likewise  Gf.  Its  order 
must  be  divisible  by  g'  and  be  a  divisor  of  ng'  (§§  10,  28). 

EXERCISES 

1.  Construct  a  group  of  order  3  of  linear  transformations  of 
determinant  unity  whose  collineation  group  is  (1,  1),  (1,  «),  (1,  <*?); 
o»3  =  l. 

2.  Construct  the  group  of  order  12  of  transformations  of  deter 
minant  unity  whose  collineation  group  is 

r-i   11     ro   n     r   o   n     r-i    n     r-i   01 

"•"•I  o  J;  L  oj'  U  ij-  U  0}  l-rij: 

(As  a  linear  fractional  group,  this  group  has  the  form 

y=y',  -y'+i,  W,  i/(-V 


13.  Change  of  variables.  For  certain  purposes  it  is 
of  great  convenience  to  introduce  new  variables  which  are 
linear  functions  of  the  old.  To  illustrate  the  theory  let 
us  consider  the  transformation  (1),  §  1: 

S:       x  —  x'  cos  0  —  y'  sin  0,      y  =  xf  sin  0+y'  cos  0. 
We  shall  now  introduce  new  variables  X,  Y,  where 
(8)  X  =  x+iy,  Y  =x-iy         (i=l/^I), 

and  correspondingly 

/0/\  ~Yf  —  /r'J-Vi/  V  —  T'       <iiif 

(o  )  A  —  x  -\-iy  y  i   —x  — ly  . 

The  transformation  S  becomes 

S':  X=ei&X',  Y=e-ieY', 

a  result  that  can  be  expressed  symbolically  in  terms  of 
S  and  the  change  of  variables  (8),  which  we  shall  regard 
as  a  linear  transformation. 


16  FINITE  COLLINEATION  GROUPS 

Solving  (8)  for  x,  y  we  obtain 
T:  x=(X+Y)/2,     y=(X-Y)/2i. 

Consider  an  arbitrary  function  f(x,  y).     The  trans 
formation  S  changes  /  into  the  function 

f(x'  cos  6-y'  sin  0,  x'  sin  6+y'  cos  0), 

which,  expressed  in  terms  of  the  variables  X',  Y',  is  the 
function 

X'+Y'            X'-Y' .     ,  X'+Y'   .       .X'-Y'        A 
—^—  cos0 -r-  sm0,  — - — sm0+     2i     cos0). 

On  the  other  hand,  we  may  at  the  outset  express  /  as  a 
function  of  the  new  variables:  f((X+Y)/2,  (X-Y)/2i), 
and  then  transform  it  by  means  of  S': 

ei9X'+e~i9Y'      ei9X'-e-i9Y'\ 
2  '  2i          )' 

Symbolically  stated,  the  operator  ST  is  equivalent  to  the 
operator  TS': 

(f)ST=(f)TSf. 
Hence, 


We  formulate  this  result  as  follows  for  the  general  case  : 

THEOREM  4.  Having  given  a  set  of  linear  transforma 
tions  A,  B,  .  .  and  a  function  f,  involving  n  variables  Xi, 
.  .  ,  xn,  we  may  introduce  new  variables  by  means  of  a 
linear  transformation 

T:     Xft  =  feiXi+fe2X2+    -    •   +fenXn     (&=!,  2,   .    .   ,  n), 


changing  f  into  a  function  of  Xi,  .  .  ,  Xn.  As  linear 
transformations  in  the  new  variables,  A,  B,  .  .  will  take 
theformsT~lAT,  T~1BT,  .  .  . 


PROPERTIES  OF  LINEAR  GROUPS 


17 


We  note  that  if  AB  =  C,  then  (T-1AT)(T~1BT) 
=  T~1CT.  Hence,  if  A,  B,  .  .  form  a  group,  so  do 
T~1AT,  t-^BT,  .  .  ,  and  the  two  groups  are  simply 
isomorphic  (§  32). 

14.  Transitive  and  intransitive  groups.  Groups  in 
twq,  three,  and  four  variables  whose  transformations  have 
respectively  the  following  forms: 


a  0  0 
0  b  c 
Ode 


p  q  0  0 

r  s  0  0 
0  0  t  u 
.0  0  v  w 


p  0  0  0 

0  a  b  c 

0  d  e  f 

-0  g  h  j. 


are  said  to  be  intransitive. 

It  may  happen  that  a  group  can  be  made  to  assume  the 
intransitive  form  by  a  suitable  change  of  variables,  though 
it  does  not  possess  this  form  initially.  Consider,  for 
example,  a  group  G  in  four  variables  x\,  .  .  ,  x4  whose 
transformations  are  all  of  the  following  type: 


a  b  e  f 

b  a  f  e 

h  g  c  d 

g  h  d  c] 


The  introduction  of  new  variables  y\,  y2,  z\y  z2,  where 
yi  =  Xi+x2,  y2  =  x3+Xi;  Zi  =  xi-x2,  z2  =  xz-x4,  will  change 
these  matrices  into  the  type  C\  above.  That  is,  G  has 
the  intransitive  form  when  written  as  a  group  in  the 
variables  yi,  y2,  2i,  z*. 

DEFINITION.  If  the  n  variables  of  a  group  G  can  be 
separated  into  two  or  more  sets  (either  directly  or  after 
a  suitable  change  of  variables),  such  that  the  variables  of 


18  FINITE  COLLINEATION  GROUPS 

any  one  set  are  transformed  by  all  the  transformations 
of  G  into  linear  functions  of  the  variables  of  that  set  only, 
we  say  that  G  is  intransitive.  If  such  a  division  is  not 
possible,  the  group  is  transitive.  The  different  sets  into 
which  the  variables  of  an  intransitive  group  may  be 
separated  (as  the  sets  (y1}  y2)  and  (zi,  z2)  of  the  group 
above)  are  called  its  sets  of  intransitivity. 

EXERCISES 

1.  Prove  that  a  change  of  variables  will  not  alter  the  form  of 
a  similarity-transformation. 

2.  Prove  that  ^'  =  [Q     °]    can  be  obtained  from  A  =  fa    £] 

by  a  suitable  change  of  variables  if  a *6,  and  find  the  correspond 
ing  transformation  T.  (Hint:  The  condition  T~1AT  =  A'  gives 
AT  =  TA'.  Multiply  out  and  determine  the  elements  in  the  matrix 
of  T  from  the  four  resulting  equations.) 

3.  Prove  that  if  in  type  Ci,  t  =  p,  u  =  q,  v=r  and  w  =  s,  then  the 
two  sets  of  intransitivity  can  be  chosen  in  an  infinite  number  of 
ways. 


HERMITIAN   INVARIANT  AND   REDUCTIBILITY   OF  LINEAR 
GROUPS,    §§  15-20 

15.  Conjugate-imaginary  groups.  Let  G=(A,  B, 
C,  .  .  )  be  a  linear  group  in  the  variables  x\,  .  .  ,  xn, 
and  assume  that  the  elements  in  the  matrices  of  the 
various  transformations  are  not  all  real.  We  may  then 
separate  the  real  and  imaginary  parts  in  both  variables 
and  coefficients;  and  by  passing  to  the  conjugate- 
imaginary  values  we  evidently  obtain  a  new  group 
G=(A,  B,  C,  .  .  )  in  the  variables  x\,  .  .  ,  xn  (we  shall 
denote  the  conjugate-imaginary  of  a  quantity  w  by  w), 
simply  isomorphic  with  G  (§32).  For,  if  AB  =  C,  it 
follows  that  AB  =  C.  We  shall  call  either  group  the 
conjugate-imaginary  of  the  other. 


PROPERTIES  OF  LINEAR  GROUPS  19 

16.  Hermitian  form.     The  expression 


k=l    1=1 

subject  to  the  condition  that  it  vanishes  only  if 
Xl  =  x2  =  .  .  =  xn  =  0,  and  is  real  and  positive  for  all  other 
sets  of  values  that  we  may  assign  to  these  variables,  is 
called  a  positive-definite  Hermitian  form  in  the  n  variables 
xlf  .  j  xnj  or  simply  Hermitian  form.  For  instance,  the 


expression   XiXi+3x2X2+(l-}-i)xiX2+(l  —  i)x2Xi  is  a   Her 
mitian  form  in  the  variables  x\,  x2. 

THEOREM  5.     A    positive-definite   Hermitian  form   J 
in  the  variables  x\,   .    .   ,  xn  may  be  reduced  to  the  form 

2/12/1  +  2/22/2+     •      •     +  2/n2/n 

by  a  change  of  variables  of  the  following  type: 
2/2  = 


Proof.  —  Arranging  J  according  to  xn  and  xn  we  have 

J  =  Jn  =  QnnXnXn  +  ^nXn  _  i  +  XnXn  _  i  +  X, 

where  Xn-i  represents  a  linear  function  of  xn-\,   .    .   ,  x\. 
The  coefficient  qnn  is  real  and  positive,  since  it  is  the 
value  of  J  obtained  by  putting 

xn=l,    xn-}  =  xn-2=   .    •    =zi  =  0. 
Accordingly,  if  we  put 

+  l/g^»=+l/^  =  pnn,  and  Xn_i  =  pnn7n_i, 
we  have 

Jn=(pnnXn+  Yn-l)  (pnnXn-l+  ?n-l)  +X  -  Y  n.,Y  n^ 


20  FINITE  COLLINEATION  GROUPS 

where  Jn-\  is  a  Hermitian  form  in  n—  1  variables  xn-i, 
.  .  ,  Xi.  For,  it  is  of  the  required  type  and  is  the  value 
of  Jn  obtained  by  subjecting  the  variables  to  the  single 
condition  xn  =  —  Yn-i/pnn.  It  is  therefore  real  and 
positive  unless  xn-\=  .  .  =  Xi  =  0. 

We  now  arrange  Jn_x  according  to  xn-i  and  zn_i,  and 
proceed  as  above.     We  find 


where  yn-i  is  a  linear  function  of  zn_i,  zn_2,   .    .  ,  *i. 
Continuing  thus,  we  finally  prove  the  theorem. 

17.  Lemma.  //  G=(Tlf  T2  .  .  .  ,  Tg)  is  a  linear 
group,  and  f  any  function  of  the  variables  of  the  group,  then 
the  function 

0)  7=(/)r,+(/)r,+  .  .  +(j)T, 

is  either  identically  zero  or  is  an  "invariant"  of  G;  that  is, 
it  is  transformed  into  itself  by  every  transformation  of  G. 
Proof.  —  We  have 

(10)       (^k=(f)T1Tk+(f)TtTk+   .    .   +  (f)T0Tk 
=  (/)«+(/)«+   •    .   +(f!M,  say. 


But,  T[,  T£,  .  .  ,  TO  are  the  transformations  Ti,  T2,  .  .  , 
Tg  over  again  in  some  order,  since  they  all  belong  to  G 
(§  7)  and  are  all  distinct  (§  5;  cf.  Exercise  3,  §  27).  It 
follows  that  the  last  sum  of  (10)  is  equal  to  the  right-hand 
member  of  (9);  i.e.,  I=(I)Tk. 

18.  Invariant  Hermitian  form.  We  say  that  a 
Hermitian  form  J  is  invariant  under  a  group  G,  or  that  J 
is  n.  J1p.rmH.ian  invariant  of  G,  when  J  is  transformed  into 
itself  by  the  transformations  of  the  intransitive^  group 
made  up  of  G  and  its  conjugate-imaginary  group  G. 


PROPERTIES  OF  LINEAR  GROUPS  21 

THEOREM  6.  There  is  always  a  Hermitian  invariant 
J  of  a  given  linear  group  G  in  n  variables* 

Proof.  —  Let  the  transformations  of  the  group  made  up 
of  G  and  G  be  denoted  by  Tit  T2,  .  .  ,  T0,  and  let  / 
represent  the  function  XiXi+xzX2-\-  .  .  +xnxn.  Then 
the  function 

.    .   +  (I)Tg 


is  the  required  Hermitian  invariant. 

First,  J  is  a  Hermitian  form  in  the  variables  x\,  .  .  , 
xn.  For  every  term  (I)Tk  is  the  sum  of  n  expressions 
(xaxa)Tk  =  XsXs,  each  of  which  is  the  product  of  two 
conjugate-imaginary  quantities.  The  function  J  is  there 
fore  real  and  non-negative,  and  cannot  vanish  unless 
every  term  (I)Tk  vanishes.  But  if  Ti  represents  the 
identity,  (I)Ti  =  I  and  does  not  vanish  unless  every 
variable  x\,  .  .  ,  xn  vanishes.  This  is  therefore  also  the 
case  with  J. 

Second,  J  is  transformed  into  itself  by  TI,  .  .  ,  Tn, 
by  the  lemma,  §  17. 

By  aid  of  the  theorems  5  and  6  we  derive  the 

COROLLARY.  Such  variables  x\,  .  .  ,  xn  may  be 
selected  for  a  linear  group  G  that  the  function 


is  a  Hermitian  invariant  of  G. 


19.  Unitary  transformations.  A  linear  transformation 
A  =  [ast]  whose  coefficients  satisfy  the  following  relations  : 

(11)    aikflih+o2ifl2k+   •    •   +ankank=l 

(fc  =  l,2,   .    -   ,  n), 

*  This  theorem  was  proved  for  n  =3  by  Picard  and  Valentiner  (1887, 
1889),  and  for  any  n  by  Puchs,  Loewy,  and  Moore  (189G).  See  Encyclo 
pedic  der  mathematischen  Wissenschaften,  Leipzig,  1898-1904,  Bd.  I,  1, 
p.  532. 


22 

FINITE  COLLINEATION  GROUPS 

(11')  t 

ti*0iz+02*o2z  +    .    .    +o«*an/=0 

(12)    a 

'*io*i+  0*2^*2+    -    .    -\-aknakn=l 

(12')  a 
is  said 

(k,l  =  l,2,  .   .-, 
to  have  the  unitary  form. 

n;  fc=H), 

The  variables  of  a  group  being  selected  so  that  its 
Hermitian  invariant  is  XiX\-\-  .  .  +xnxn,  we  readily 
find  that  the  corollary  of  §  18  is  tantamount  to  the  follow 
ing  statement :  such  variables  may  be  selected  for  a  linear 
group  that  its  transformations  all  have  the  unitary  form. 

The  equations  (11),  (11')  are  deduced  directly;  and  the  equa 
tions  (12),  (12')  by  operating  on  the  Hermitian  form  by  A~l,  as 
given  below. 

The  inverse  of  a  transformation  in  unitary  form  can  be 
written  down  at  once: 

A-1:  xk  =  aikx'1+a2kx2+   .    .   +ankxll     (k  =  l,  2,   .    .   ,  n). 

For,  the  condition  A~1A  =  (l,  1,   .    .   ,  1)  leads  to  the 
equations  (11),  (!!')• 

20.  Reducible  and  irreducible  groups.  A  linear 
group  in  n  variables  is  said  to  be  reducible  when,  after  a 
suitable  choice  of  variables  x\,  .  .  ,  xn,  a  certain  number 
of  these  (say  Xi,  z2,  .  .  ,  xm;  m<n)  are  transformed 
into  linear  functions  of  themselves  by  the  transforma 
tions  of  the  group.  (Thus,  a  group  in  three  variables 
Xi,  xz,  z3,  whose  matrices  have  the  form 


PROPERTIES  OF  LINEAR  GROUPS  23 

is  reducible.)     We  shall  say  that  the  m  variables  x\t   .    .   , 
xm  constitute  a  reduced  set  of  the  group. 

An  irreducible  group  is  a  group  in  which  no  such  choice 
of  variables  is  possible. 

THEOREM  7.  A  reducible  group  G  is  intransitive,  and  a 
reduced  set  becomes  a  set  of  intransitivity. 

Applied  to  the  reducible  group  in  three  variables  indicated 
above,  the  theorem  asserts  that  new  variables  may  be  introduced 
such  that  the  matrices  are  of  type  B,  §  14,  and  that  the  original 
variables  x\,  x2  will  form  a  set  of  intransitivity  of  the  new. 

Proof. — A  Hermitian  invariant  of  G  may  be  reduced 
to  the  form  y\y\-\-  .  .  -\-ynyn  by  the  change  of  variables 
specified  in  §  16.  The  group  G  will  still  have  the  typical 
form  of  a  reducible  group,  whose  matrices  we  shall  write 
symbolically 


\A'    0 

LA"  A'"\ ' 


and  it  remains  for  us  to  prove  that  the  elements  in  A  "  are 
all  zero. 

For  this  purpose  we  write  down  as  many  of  the  equa 
tions  (12)  and  as  many  of  the  equations  (11)  as  contain 
elements  of  A'";  namely  the  last  n-m  in  each  case.  If 
we  then  subtract  the  sum  of  the  latter  equations  from  the 
sum  of  the  former,  there  results  an  equation  2as<as<  =  0, 
in  which  the  left-hand  member  contains  one  term  for 
each  of  the  elements  of  A".  Now,  each  of  these  terms 
is  real  and  non-negative;  consequently  the  sum  2<astast 
cannot  vanish  unless  every  number  as(  =  0.  That  is,  the 
elements  of  A"  are  all  zero,  and  the  theorem  is  proved. 

The  validity  of  the  proof  is  evidently  not  impaired  by 
assuming  the  given  reducible  group  G  to  be  a  subgroup  of 
a  larger  group  H,  reducible  or  irreducible.  Furthermore, 
the  final  intransitive  group  G  is  composed  of  unitary 


24  FINITE  COLLINEATION  GROUPS 

transformations,  by  the  process  of  proof;  and  this  result 
is  equally  true  of  H  (let  the  initial  Hermitian  invariant 
used  in  the  proof  be  the  Hermitian  invariant  of  H) .  Thus, 
with  slight  modifications  of  the  above  proof  as  to  detail, 
which  will  be  left  as  an  exercise  for  the  student,  we  obtain 
the  following  important 

THEOREM  8.  Having  given  a  linear  group  H  contain 
ing  an  intransitive  subgroup  G,  we  may  choose  such  variables 
that  G  appears  in  intransitive  form  (cf.  types  A-C,  §  14), 
and  that  at  the  same  time  the  transformations  of  H  are  all 
unitary. 

EXERCISES 

1.  Show  that  a  unitary  transformation  in  two  variables  and  of 
determinant  unity  has  the  form 

pp+qq  =  l  . 

2.  Prove  by  the  method  of  §  18  that  if  all  the  elements  in  the 
matrices  of  G  are  real,  then  there  is  a  quadratic  function  of  the 
variables  which  is  invariant  under  G. 

3.  Find  the  most  general  type  of  a  Hermitian  invariant  of  a 
linear  group  which  contains  a  transformation  in  canonical  form 
whose  multipliers  are  all  distinct,  as  (1,  w,  w2);   «3  =  1. 

4.  Prove  that  if  a  group  G  possesses  a  Hermitian  invariant 
which  does  not  contain  all  the  variables  of  G,  then  this  group  is 
intransitive. 

5.  If  there  exists  a  linear  function  of  the  variables  of  a  group 
G  which  is  invariant  under  G,  then  this  group  is  intransitive. 

In  the  case  of  a  substitution  group  (chap,  ii)  written  as  a  linear 
group  (§1)  there  is  such  a  function,  namely  the  sum  of  the  letters 
of  substitution.  Hence  this  group  is  always  intransitive. 

CANONICAL  FORM   OF  A   LINEAR  TRANSFORMATION  AND   OF 
ABELIAN   GROUPS,    §§  21-22 

21.  Theorem  9.  A  linear  transformation  of  finite 
order  will  assume  the  canonical  form  by  a  suitable  choice 
of  variables. 


PROPERTIES  OF  LINEAR  GROUPS  25 

Proof.  —  Let  the  transformation  be  A  =  [ast\.  We  can 
determine  a  linear  function  which  is  transformed  into  a 
certain  constant  multiple  of  itself  by  A,  say  (y\)A=Oy^ 
where  y\  =  biXi+  .  .  +bnxn.  To  obtain  the  necessary 
conditions  we  equate  the  coefficients  of  the  variables 
Zi,  .  .  ,  xn,  and  find  the  equations 

biait+b2flu+   .    .   +bnant  =  0bt       (*=1,  2,   .    .   ,  n), 

which  can  be  solved  for  61,   .    .   ,  bn  provided  0  is  a  root 
of  the  characteristic  equation  of  A  (cf.  §  23). 

Having  thus  determined  yi  We  change  to  new  variables 
such  that  y\  is  one  of  these.  The  group  generated  (§8) 
by  A  is  now  seen  to  be  reducible,  since 


from  which  it  follows  that  the  first  row  in  the  matrices 
of  each  of  these  transformations  is  of  the  form 

k  0  0  .    .0, 

where  k  represents  different  powers  of  0  in  the  different 
transformations.  Hence,  by  Theorem  7,  §  20,  the  group 
in  question  is  intransitive,  and  yi  constitutes  one  of  its 
sets  of  intransitivity.  Let  (yz,  .  .  ,  yn)  be  the  other 
(temporary)  set  of  intransitivity. 

The  foregoing  process  may  now  be  repeated  for  the  set 
(2/2,  •  •  ,  2/n)  in  place  of  the  original  n  variables.  A  new 
linear  function  will  be  determined  which  is  transformed 
into  a  constant  multiple  of  itself  by  A,  and  the  set 
(2/2,  •  •  ,  yn)  will  break  up  into  further  sets  of  intransi 
tivity.  Continuing  thus,  we  finally  obtain  A  in  the  desired 
canonical  form. 

22.  Canonical  form  of  abelian  group.  The  group 
consisting  of  the  different  powers  of  a  transformation  A 


26  FINITE  COLLINEATION  GROUPS 

is  an  example  of  a  type  of  group  called  abelian;  namely 
a  group  in  which  all  the  operators  are  commutative  (§4); 
i.e.,  if  A  and  B  are  two  operators  of  the  group,  then 
AB  =  BA. 

THEOREM  10.  In  any  given  abelian  group  K  of  linear 
transformations,  such  new  variables  may  be  introduced 
that  all  the  transformations  of  K  will  simultaneously  have 
the  canonical  form. 

Proof.  —  If  the  group  contains  only  similarity- 
transformations  (§2)  the  theorem  is  self-evident.  Hence 
we  assume  in  K  a  transformation  S  which  is  not  a 
similarity-transformation.  Let  the  variables  of  the  group 
be  chosen  such  that  S  appears  in  the  canonical  form  (§  21)  : 

s=(«,  ..,*)"; 

The  multipliers  of  S  may  not  all  be  distinct.  Sup 
pose  that  m  of  them  are  equal  (say  to  a),  and  differ  in 
value  from  all  the  others;  we  shall  then  show  that  K 
transforms  the  corresponding  variables  into  linear  func 
tions  of  themselves  and  is  therefore  intransitive  (§  20). 

Let  /  be  any  linear  function  of  the  m  variables  in  ques 
tion,  say  Xi,  Xz,  .  .  ,  xm.  We  have  (f)S  =  af;  moreover, 
any  linear  function  F  such  that  (F)S  =  aF  can  evidently 
not  contain  any  of  the  variables  xm+ij  .  .  ,  xn. 

Let  now  T  be  any  transformation  of  K.  We  have 
ST=TS,  and  if  we  put  (f)T=f  we  get 


That  is,  the  function  /',  which  is  linear  in  the  variables 
of  the  group,  must  be  a  function  of  x\,  .  .  ,  xm  only, 
by  what  has  just  been  said.  It  follows  that  K  is  intransi 
tive. 


PROPERTIES  OF  LINEAR  GROUPS 


27 


If  we  now  confine  our  attention  to  one  of  its  sets  of 
intransitivity,  we  may  apply  anew  the  process  above  to 
that  set.  This  will,  therefore,  break  up  into  further  sets 
of  intransitivity.  Continuing  thus,  the  ultimate  sets 
of  intransitivity  will  contain  one  variable  each,  and  the 
theorem  is  proved. 

We  shall  often  say:  "let  a  (given)  transformation  (or  group) 
be  written  in  canonical  form"  instead  of  "let  the  variables  be  so 
chosen  that  a  (given)  transformation  (or  group)  will  appear  in  the 
canonical  form." 

From  the  theorems  8  and  10  we  deduce  the 

COROLLARY.  Such  variables  may  be  selected  for  the 
variables  of  a  linear  group  G  that  a  given  abelian  subgroup 
of  G  is  written  in  the  canonical  form,  and  that  at  the  same 
time  the  transformations  of  G  are  all  unitary. 

23.  Characteristic  and  characteristic  equation.     If  we 

add  —  0  to  each  of  the  elements  in  the  principal  diagonal 
of  the  matrix  of  a  linear  transformation  A  =  [ast]  and 
equate  to  zero  the  resulting  determinant,  we  have  an 
equation  in  0  which  is  called  the  characteristic  equation  of  A: 


(13) 


On  —  ^    Oi2 
021  022  — 


,-e 


=  0 


THEOREM  11.  If  T  and  A  are  linear  transformations, 
the  roots  of  the  characteristic  equation  of  A  are  the  same  as 
those  of  T~l AT. 

Proof.— Put  T-1AT  =  B  =  [bst],  whose  characteristic 
equation  is 

bii  —  0.    .  bin 
(14)  =0. 

bni        .    .  bnn-0 


28  FINITE  COLLINEATION  GROUPS 

Regarding  0  as  a  variable  temporarily,  we  shall  look 
upon  the  left-hand  members  of  (13)  and  (14)  as  the 
matrices  of  linear  transformations  which  we  shall 
write  symbolically  (A—  S)  and  (B  —  S),  where  S  repre 
sents  the  similarity-transformation  (0,  .  .  ,  0).  Then, 
since  T~1AT  =  B  and  T~1ST  =  S)  we  readily  find  that 
T~l(A -S)T=  (B-S).  Hence,  if  the  determinants  of  T, 
(A—S)  and  (B  —  S)  are  denoted  by  t,  a,  and  b  respec 
tively,  we  have  (cf.  Exercise  3,  §  11)  t'1  at  =  b,  giving  a  =  b. 
Accordingly,  the  coefficients  of  the  various  powers  of  6 
in  a  and  b  are  equal,  and  the  theorem  follows. 

The  sum  of  the  characteristic  roots  of  A  is  called  the 
characteristic  of  A.  It  is  equal  to  the  sum  of  the  elements 
in  the  principal  diagonal  of  A,  namely  011+022+  .  •  +«nn- 

EXERCISES 

1.  Find  the  characteristic  roots  of  a  transformation  written 
in  canonical  form. 

2.  Prove  that  the  characteristic  roots  of  a  linear  transformation 
of  finite  order  are  roots  of  unity  (cf.  §§  4,  6,  21,  23,  133). 

r  i  °i 

3.  Can   the  variables  be  so  changed  that  S  =  will 
assume  the  canonical  form  ? 


CHAPTER  II 

GROUPS  OF  OPERATORS  AND   SUBSTITUTION   GROUPS 
A.    GROUPS  OF  OPERATORS 

24.  Introduction.  The  notion  of  a  group  was  intro 
duced  in  §  7.  While  this  term  has  been  associated 
hitherto  with  linear  transformations  only,  there  are  so 
many  important  properties  of  groups  of  operators  which 
are  independent  of  their  mode  of  representation,  that  it 
seems  best  to  study  such  properties  apart  from  the  form 
that  these  groups  take.  The  usefulness  of  the  results 
derived  will  in  this  way  not  be  limited  to  the  realm  of 
linear  groups. 

In  addition  we  need  a  certain  amount  of  knowledge 
of  substitution  groups  for  the  development  of  linear 
groups.  However,  beyond  a  general  introduction  to  the 
theory  of  groups  of  operators  and  substitution  groups,  only 
such  additional  theorems  in  either  field  as  are  needed  for 
this  development  will  be  given  here.  For  a  detailed 
account  in  the  English  language  of  abstract  and  substitu 
tion  groups,  the  reader  may  with  profit  consult  the  follow 
ing  books: 

E.  Netto,  The  theory  of  Substitutions  and  Its  Applications  to 
Algebra  (tr.  by  F.  N.  Cole).  Ann  Arbor,  Mich.:  Inland 
Press,  1892. 

W.  Burnside,  Theory  of  Groups  of  Finite  Order.  2d  ed.  Cam 
bridge  University  Press,  1911. 

H.  Hilton,  An  Introduction  to  the  Theory  of  Groups  of  Finite 
Order.  Oxford,  1908. 

G.  A.  Miller,  H.  F.  Blichfeldt,  and  L.  E.  Dickson,  Theory  and 
Applications  of  Finite  Groups.  New  York:  John  Wiley 
&  Sons,  1916. 

29 


30  FINITE  COLLINEATION  GROUPS 

OPERATORS   AND    GROUPS   OF   OPERATORS,    §§  25~28 

25.  Operators.*  We  postulate  a  certain  class  O  of 
objects  called  operators  having  the  following  properties: 

1°.  If  A  and  B  are  any  two  operators,  then  either  A 
and  B  are  equal  (A=B)  or  distinct  (A^B). 

That  is,  the  operators  are  so  defined  that  it  shall  be  possible 
for  us  in  any  case  to  determine  whether  two  given  operators  are 
equal  or  not.  Thus,  if  O  is  the  class  of  all  linear  transformations 
in  two  variables,  two  operators  A  and  B,  defined  by  their  matrices 


3 

Lc    dJ 


-C 

are  equal  when  the  following  (ordinary)  equations  are  satisfied: 
a  =  p,  b  =  q,  c  =  r,  d=s;  if  O  is  the  class  of  all  collineations  in  two 
variables,  the, two  corresponding  operators  are  equal  if  a  number  k 
can  be  found  such  that  the  equations  a  =  pk,  b  —  qk,  c  =  rk,  d  =  sk 
are  satisfied. 

2°.  Whether  A  =B  or  A  4=#,  there  is  a  unique 
operator  C  called  the  product  of  A  and  B;  in  symbols 
AB  =  C. 

Here  we  assume  that  a  certain  rule  for  forming  the  product 
AB  (cf .  §  3)  is  given,  producing  an  operator  C  in  O,  which  is 
"unique"  in  the  sense  that  if  more  than  one  operator  results  from 
the  rule,  then  any  two  such  operators  are  equal  (cf.  §  9). 

3°.  The  associative  law  holds  for  a  product  of  three 
operators:  (AB)C  =  A(BC}\  that  is,  if  AB  =  S,  BC  =  T, 


4°.  There  is  a  unique  operator  E  called  the  identity 
and  having  the  property  that,  for  every  operator  A,  we 
have  AE  =  A  and  EA  =A. 

5°.  The  operators  are  reversible  in  0;  that  is,  to  every 
operator  A  there  corresponds  a  unique  operator  in  O 

*  The  development  of  chap,  i  is  followed  closely  in  §§  25-27.  For  a 
bibliography  and  discussion  of  various  definitions  of  abstract  groups  con 
sult  E.  V.  Huntington,  Transactions  of  the  American  Mathematical  Society, 
VI  (1905),  181  ft*.  The  postulates  4°  and  5°  above  demand  more  than  is 
logically  necessary  (cf.  L.  E.  Dickson,  Transactions  of  the  American  Mathe 
matical  Society,  ibid.,  p.  199). 


GROUPS  OF  OPERATORS  31 

called  the  inverse  of  A  and  denoted  by  A"1,  such  that  we 


In  4°  and  5°  the  word  "unique"  is  interpreted  as  in  2°. 

6°.  The  following  relations  of  equality  obtain,  as  in  the 
case  of  ordinary  (numerical)  equality:  (a)  A=A;  (6)  if 
A=B,  then  B  =  A;  (c)  if  A=B  and  B  =  C,  then  A  =  C; 
(d)  if  A  =  B  and  C  =  D,  then  AC  =  BD. 

Remark.  —  Our  conception  of  the  class  of  operators  O  involves 
necessarily  another  class  F,  composed  of  subjects  of  operation.  To 
give  some  illustrations:  (a)  let  F  represent  all  polynomials  in  n 
variables  and  O  all  linear  transformations  in  those  variables;  (6)  let 
F  represent  all  points  in  the  plane  and  O  all  rotations  in  that  plane 
around  a  given  point,  accompanied  or  not  by  inversions  with  respect 
to  the  given  point;  (c)  let  F  represent  n  points  on  a  line  and  O  the 
different  permutations  of  those  points.  However,  in  the  present 
chapter  the  class  F  is  practically  never  referred  to. 

The  product  AB  is  not  necessarily  the  same  as  the 
product  BA.  If  the  two  products  are  equal  (AB  =  BA), 
we  say  that  the  operators  A  and  B  are  commutative.  A 
continued  product  of  any  number  of  operators  A,  B,  C, 
D,  .  .  may  be  obtained  by  taking  the  product  of  two 
of  them  (say  AB),  then  the  product  of  this  product  and 
an  operator,  etc.,  giving  say  (((AB)C)D)  .  .  .  .  By 
3°  it  follows  that  the  factors  in  the  final  product  may  be 
reassociated  in  any  manner  (as  (AB)(CD)  .  .  ),  so  long 
as  their  order  in  the  product  is  not  disturbed. 

In  the  future  we  shall  often  say  "a  (set,  group,  class) 
G  contains  an  operator  S,"  "S  is  found  among  the 
operators  of  G,"  "S  is  an  operator  of  G,"  or  simply  "S 
belongs  to  G,"  instead  of  "S  is  equal  to  one  of  the  operators 
of  G.)} 

EXERCISES 

1.  Prove  that  the  inverse  of  the  identity  is  the  identity  (i.e., 
E~l=E),  and  that  the  inverse  of  A~l  is  A  (i.e.,  A~l  A=E). 

(Hint:  let  A'  be  the  inverse  of  A~l,  then  A~1A'  =  E.  Now 
apply  3°  to  the  left-hand  member  of  A(A~1A')=AE.) 


32  FINITE  COLLINEATION  GROUPS 

2.  Prove  that,  if  AB  =  AC,  or  if  BA  =CA,  then  B  =  C. 

3.  Prove  that  (S~1AS)  (S-1BS)=S~1(AB)S.    , 

4.  Let  the  operators  AI,  A2,   .    .   of  a  class  O  be  represented 
by  the  symbols  (x\,  yi),  (x2,  y2),   .    .   ,  under  the  condition  that  two 
operators  are  equal   (Ai  =  A2)   only  if  the  two  equations  Xi  =  x2, 
t/i  =  i/2  are  satisfied.     Now  if  the  product  A  \A2  is  expressed  by  the 
formula 

2,  2/2)  =  (x',  y'), 


where  x'  =  XiX2  —  yiy2,  y'  =  x\yi  +y\x<L,  prove  that  the  conditions  1°  —  6° 
are  fulfilled,  and  find  the  identity  and  the  inverse  of  A\.  (To  give 
an  example  of  an  operator  of  this  type:  let  operating  by  A!  con 
sist  in  multiplying  by  the  complex  number  Xi+V  —  li/i.) 

If  the  product  is  (x'y  y'}  =  (xiX2,  Xiy2),  prove  that  1°,  2°,  3°,  6° 
are  fulfilled. 

5.  Prove  that  the  inverse  of  the  operator  AB  is  B~1A~1. 

26.  Power  and  order  of  an  operator.  As  in  §4,  we 
write  A2  for  A  A,  A3  for  A  (A2)  or  (A2)  A,  .  .  ,  Am  for 
A(Am~l)  or  (Am~1)A)  and  call  these  products  the  2d, 
3d,  .  .  ,  mth  powers  of  A.  We  also  write  A~m  for 
(Am)~l,  and  if  we  put  A°  =  E  we  have 


where  m  and  n  are  positive  or  negative  integers  or  zero. 

If  a  certain  power  of  the  operator  A  equals  the  identity, 
then  A  is  of  finite  order;  and  the  least  positive  integer  m 
such  that  Am  =  E  is  called  the  order  of  A.  We  shall  prove 
the  following  propositions: 

(a)  If  An  =  E,  then  n  is  a  multiple  of  m.     For  if  not, 
let  r  be  the  remainder  when  n  is  divided  by  m,  so  that 
n  =  mq+r,  and  r<m.     Then  An  =  Am«+r  =  Am«Ar  =  E'*Ar 
=  Ar  =  E,  which  is  contrary  to  the  hypothesis  that  Am  is 
the  least  power  of  A  which  equals  E. 

(b)  The  order  of  Ak  is  m/d,  where  d  is  the  highest  com 
mon  factor  of  m  and  k.     For,  the  order  of  Ak  is  the  least 
positive  integer  t  for  which  E=  (Afc)t  =  Akt;  i.e.,  for  which 
kt  is  a  multiple  of  m,  by  (a)  .     Hence,  kt/m  =  an  integer  ;  and 


GROUPS  OF  OPERATORS  33 

canceling  the  common  factor  d  from  k  and  m,  the  remaining 
factor  of  m  (namely  m/d)  must  divide  t;  that  is,  t  =  m/d. 

27.  Finite  groups.  Generators.  We  shall  now  take 
up  the  study  of  sets  of  operators  called  finite  groups. 

DEFINITION.  A  set  of  g  distinct  operators  Si,  S2, 
.  .  ,  SginO  form  a  group  G  of  order  g  if  the  product  of  any 
two  of  them,  whether  equal  or  distinct,  is  an  operator  of  the 
set.  We  write  G=  (&,  S2,  .  .  ,  Sg). 

If  H=(Ti,  T%,  .  .  ,  Tg)  is  another  group  containing 
the  same  operators  as  G,  we  write  G  =  H.  We  are,  how 
ever,  not  to  infer  that  the  operators  are  arranged  in  the 
same  order;  that  is,  G  =  H  does  not  necessarily  imply 
Si=Ti,  .  .  ,  Sg=Tg.  If  there  is  at  least  one  operator 
in  G  not  found  in  H ,  or  vice  versa,  we  shall  say  that  the 
two  groups  are  distinct  (G^pH). 

Example  1. — Let  O  consist  of  all  rotations  of  a  sphere  around 
its  center.  The  three  rotations,  each  of  180°,  around  each  of  three 
mutually  perpendicular  diameters,  together  with  the  identity  (no 
rotation),  form  a  group  of  order  4. 

Example  2. — Let  O  consist  of  the  different  permutations  of  four 
points  a,  b,  c,  d  on  a  line.  The  four  permutations  (§  40): 

/a  b  c  d\      /ab  c  d\      /a  b  c  d\      /a  b  c  d\ 
(abed)'    \badc)'    \cdab)'    \dcba) 

form  a  group  of  order  4. 

We  find,  as  in  §  8,  that  the  identity  belongs  to  G; 
that  each  operator  of  G  is  of  finite  order;  and  that  the 
inverse  of  each  operator  of  G  is  an  operator  of  G. 

We  shall,  obviously,  exclude  from  the  concept  "group"  the 
(trivial)  group  consisting  of  the  single  operator  E. 

GENERATORS.     A  set  of  operators  A,   B,   .    .  of  G 
having  the  property  that  every  operator  S  of  G  is  expres 
sible  as  a  product  of  the  operators  of  the  set: 
S=    .    .  AaBb  .    .  AcBd  ...   ,  is  called  a  set  of  generators 


34  FINITE  COLLINEATION  GROUPS 

of  G.     Thus,  any  two  of  the  rotations  of  180°  in  the  group 
of  Example  1  above  will  generate  that  group. 

EXERCISES 

1.  Prove  that  a  set  of  operators  will  form  a  group  H  if  it  be 
known  that  they  belong  to  a  given  group  and  that  the  product  of 
any  two  of  them,  alike  or  distinct,  is  an  operator  of  the  set  H. 

2.  Show  that  if  A,  B,   .    .   belong  to  a  group  G,  then  any 
product  of  three  or  more  of  these  operators,  as  for  instance  A~1BA, 
is  again  an  operator  of  G. 

3.  Let  Si,  Sz,   .    .   ,  Sg  be  the  different  operators  of  a  group  G, 
and  let  S  represent  any  one  of  them.     Prove  that  the  g  operators 
obtained  by  taking  the  products  SiS,  $2$,   .    .   ,  SaS  are  all  distinct 
and  are  therefore  the  operators  of  G  over  again  in  some  order. 

4.  Find  a  set  of  two  generators  of  the  group  in  Example  2  above. 

5.  Consider  the  symbols  (xi,  yi),   .    .   of  Exercise  4,  §  25,  the 
rule  for  a  product  being  the  first  of  the  two  there  given.     Prove  that 
(0,  1)  is  a  generator  of  a  group  of  order  4.     Prove  also  that  the  two 
symbols  (-1,  0),  (-1/2,  1/3/2)  generate  a  group  of  order  6. 

Both  of  these  groups  are  abelian  (§34). 

6.  Prove  that  the  operators  common  to  two  groups  form  by 
themselves  a  group. 

28.  Subgroups.  A  group  H  of  order  h,  all  of  whose 
operators  are  found  among  thosye  of  a  group  G,  is  called 
a  subgroup  of  G.  Strictly  speaking,  H  is  not  thought  of  as 
a  subgroup  unless  h<g,  but  we  shall  here  generally  under 
stand  the  definition  in  such  a  way  that  G  is  a  subgroup  of 
itself  (h  =  g). 

THEOREM  1.  The  order  of  a  group  G  is  divisible  by 
the  order  of  a  subgroup  H  of  G:  g  =  hk.  The  quotient 
g/h  =  k  is  called  the  index  of  H. 

The  proof  is  based  on  our  arranging  the  operators  of 
G  in  the  form  of  a  rectangle  of  h  columns  and  k  rows: 

E,      Sz)  •     •     i    Sh,' 

Vz,  SzVz,  .    .  ,  ShV2; 

(i)  v 


vk,  s2vk,  ,     ,  shvk. 


GROUPS  OF  OPERATORS  35 

The  first  row  is  composed  of  the  operators  of  H;  Vz  is  an 
operator  of  G  not  found  among  those  of  the  first  row; 
Vs  is  an  operator  of  G  not  found  among  those  of  the  first 
two  rows,  etc.  New  rows  are  added  in  this  manner  until 
every  operator  of  G  has  been  accounted  for.  It  remains  for 
us  to  prove  that  the  hk  operators  obtained  are  all  distinct. 

First,  the  operators  in  any  one  row  are  distinct.  For 
the  assumption  SaVc  =  SbVc  gives  Sa  =  Sb  (Exercise  2, 
§  25),  which  is  contrary  to  the  hypothesis  that  the  first 
row  is  composed  of  the  distinct  operators  of  H. 

Second,  the  operators  from  two  different  rows  are 
distinct.  For  if,  say,  SaVc  =  SbVd,  c<d,  we  should  have 
Sb-1(SaVe)=Sb-1SbVd;  that  is,  Vd  =  S'Ve,  where  S'  = 
Sb~lSa  and  therefore  belongs  to  H,  since  S^1  and  Sa  do 
that.  But  then  Vd  would  occur  in  a  previous  row  (the 
cth),  contrary  to  hypothesis. 

It  follows  that  the  table  contains  all  the  operators  of 
G,  once  each.  The  theorem  is  therefore  proved. 

We  shall  indicate  the  rows  symbolically  by  H,  HVz, 
.  .  ,  HVk,  and  write 

G  =  H+HV2+   .    .   +HVk. 

It  is  to  be  noticed  that  the  h  operators  in  any  one  row,  except 
the  first,  do  not  form  a  group. 

COROLLARY.  The  order  of  an  operator  S  of  a  group  G 
is  a  factor  of  the  order  g.  For,  if  m  is  the  order  of  S,  the 
m  operators  E,  S,  S2,  .  .  ,  >Sm~1  form  a  subgroup  of  G. 

Remark. — Though  the  order  of  a  subgroup  of  G  is  a  factor  of 
the  order  of  G,  it  is  not  true  in  general  that  there  is  a  subgroup  whose 
order  is  any  given  factor  of  g,  unless  this  factor  is  a  power  of  a  prime 
number  (cf.  §  36,  and  Exercise  1,  §  38). 

EXERCISES 

1.  Find  the  subgroups  of  order  2  of  the  group  in  Example  1,  §  27. 

2.  Construct  a  subgroup  of  order  2  and  one  of  order  3  of  the 
collineation  group  given  in  Exercise  2,  §  12. 


36  FINITE  COLLINEATION  GROUPS 

CONJUGATE   SETS,    §§  29~31 

29.  Having  defined  groups  and  subgroups,  we  now 
introduce  the  first  of  two  important  concepts,  conjugate 
sets  and  isomorphism. 

DEFINITION.  If  A  and  S  are  operators  belonging  to  a 
group,  then  A  and  S-1AS  are  said  to  be  conjugate.  We 
also  say  that  the  first  of  these  operators  is  transformed  into 
the  last  by  S. 

The  following  propositions  (a)  and  (b)  are  easily 
verified  by  the  student  : 

(a)  There  is  an  operator  which  transforms  S~*AS  into 
A,  namely  S~*.     It  follows  that  the  relationship  expressed 
by  "conjugate"  is  reciprocal. 

(b)  If  A  and  B  are  conjugate,  and  also  B  and  C,  then 
A  and  C  are  conjugate.     (Prove  that  if  A  is  transformed 
into  B  by  S  (B  =  S~1AS),  and  B  into  C  by  T,  then  A  is 
transformed  into  C  by  ST:  C  =  (ST)-1A(ST).) 

Now  let  S  be  any  one  of  the  operators  of  the  group 
G=(E,  Szj  .  .  ,  Sg).  Consider  the  conjugates: 

(2)  E->SE=S,  S 


They  are  not  all  distinct:  all  the  operators  E,  S,  .  .  of 
G  which  are  commutative  with  S  (and  only  those) 
transform  S  into  itself.  For,  from  S  =  SalSSa  follows 
SaS=SSa  and  vice  versa.  We  therefore  select  a  repre 
sentative  of  each  distinct  conjugate  and  get  what  is 
called  a  complete  set  (or  simply  set)  of  conjugate  oper 
ators  under  G. 

(c)  A  conjugate  set  A,  B,  .  .  .  is  transformed  into 
itself  by  any  operator  S'  of  G.  For,  if  A  =  S^SSa,  B  = 
SblSSb,  •  •  are  distinct  conjugates,  so  are  Sf~lAS't 
S'~1BS',  .  .  ,  and  if  the  former  set  is  taken  from  the 
series  (2),  the  latter  must  belong  to  (2)  also,  since 
S'-l(S^SSa)S'=(SaS')-lS(SaS'),  etc.  (cf.  Exercise  5,  §  25). 


GROUPS  OF  OPERATORS  37 

| 

The  number  of  operators  in  a  conjugate  set  is  deter 
mined  by  the  following: 

THEOREM  2.  All  the  operators  E,  Sa,  Sb,  .  .  of  G 
which  are  commutative  with  S  form  a  subgroup  H  of  G, 
say  of  order  h,  and  the  conjugate  set  of  G  to  which  S  belongs 
contains  g/h  distinct  operators. 

To  prove  the  first  statement,  we  need  merely  to  show 
that  the  operator  SaSb  is  commutative  with  S  (Exercise  1, 
§  27).  This  is  done  as  follows: 

(SaSb)S  =  Sa(SbS)  =  Sa(SSb)  =  (SaS)Sb  =  (SSa)Sb  =  S(SaSb)  . 

To  prove  the  second,  consider  the  table  (1),  §  28. 
The  operators  in  the  first  line  will  all  transform  S  into 
itself,  since  they  are  here  commutative  with  S.  All  those 
in  the  second  line  will  transform  S  into  one  and  the  same 
(new)  operator  VzlSV2.  For,  SaVz  being  any  operator 
of  HVz,  we  have 


Moreover,  VzlSV2  is  distinct  from  S,  as  otherwise  V2 
would  be  commutative  with  S  and  would  therefore  belong 
to  H.  Similarly,  the  operators  of  the  third  line  all  trans 
form  S  into  a  third  operator  V^SVs,  distinct  from  S 
and  from  V^SV*  Thus,  suppose  V^SV^  V^SVz;  then 
S  would  be  commutative  with  V^V^1,  so  that  this  operator 
would  be  an  operator  in  H,  say  Sa.  Hence  Vs  =  SaV2. 
But  this  is  impossible,  since  Fa  is  not  found  among  the 
first  two  lines  of  the  table  (1).  Proceeding  thus,  we  obtain 
a  distinct  conjugate  for  each  line  of  the  table,  and  the 
theorem  is  proved. 

30.  Conjugate  subgroups.  If  we  transform  the  opera 
tors  E,  S2,  .  .  ,  Sk  of  a  subgroup  K  of  G  by  an  operator 
of  G,  say  V,  we  obtain  the  same  or  another  subgroup 


38  FINITE  COLLINEATION  GROUPS 

which  we  shall  designate  V~1KV.  For,  the  resulting  k 
operators  belong  to  G,  and  the  product  of  any  two  of 
them: 

(V-*SaV)(V-*SbV)  =  V-lSaVV-lSbV=  V~l(SaSb)V, 

is  contained  in  the  set  V~1KV)  since  SaSb  is  an  operator 
of  K. 

We  say  that  K  and  V~1KV  are  conjugate  subgroups  of 
G.  If  the  two  groups  are  equal  (K=  V~1KV;  cf.  §  27), 
we  say  that  V  is  commutative  with  K. 

The  reader  may  verify  the  propositions  corresponding 
to  (a),  (6),  and  (c),  §  29,  namely  that  V~1KV  is  trans 
formed  into  K  by  V~1,  that  two  groups  conjugate  to  a 
third  are  conjugate  to  each  other,  and  that  a  set  of  con 
jugate  subgroups  of  G  is  transformed  into  the  same  set 
by  an  operator  of  G.  Finally,  the  following  theorem  may 
be  proved  in  the  same  manner  as  the  corresponding 
theorem  above: 

THEOREM  2'.  All  the  operators  of  G  which  are  com 
mutative  with  K  (among  these  are  found  the  operators  of 
K)  form  a  subgroup  H'  of  order  h',  and  the  number  of  distinct 
subgroups  of  G  conjugate  to  K  is  g/h'. 

EXERCISES 

1.  Prove  that  the  operators  AB  and  BA  are  conjugate. 

2.  Prove  that  conjugate  operators,  or  conjugate  subgroups,  have 
the  same  order. 

3.  If  two  commutative  operators,  A  and  B,  are  transformed  by 
an  operator  S,  the  new  operators  are  also  commutative. 

Hence,  the  conjugate  of  an  abelian  group  (§  34)  is  again  an 
abelian  group. 

4.  If  H  is  the  group  whose  operators  are  commutative  with  S 
(cf.  Theorem  2),  then  T~1HT  is  the  group  whose  operators  are  com 
mutative  with  T-WT. 

5.  Prove  that  two  sets  of  conjugate  operators  have  either  no 
operators  in  common  or  are  composed  of  the  same  operators. 

Hence  show  that  the  operators  of  G  can  be  separated  into  distinct 
conjugate  sets,  the  total  number  of  whose  operators  is  equal  to  the 


GROUPS  OF  OPERATORS  39 

order  (0)  of  the  group.  At  least  one  set  contains  only  one  operator, 
namely  the  identity.  Accordingly,  if  these  sets  contain  respectively 
1,  k2,  fc3,  .  .  operators,  we  have  0  =  1+  &2+fc3+  .  .  .  Further 
more,  we  have  (Theorem  2)  k2*=g/h2,  k3  =  g/h3,  .  .  ,  so  that  finally 


31.  Invariant  operators  and  subgroups.  Simple 
groups.  We  say  that  the  operator  S  is  invariant  under 
G,  or  that  it  is  a  self  -conjugate  operator  of  G,  when  S  is 
transformed  into  itself  by  every  operator  of  G  (that  is, 
if  the  operators  (2),  §  29,  are  all  equal). 

Similarly,  we  say  that  a  subgroup  H  of  G  is  invariant 
under  G  or  is  a  self-conjugate  subgroup  of  G,  when 
H  =  S2~1HS2=  .  .  =  Sg~1HSg. 

Thus,  the  operator  S  is  invariant  under  H  in  Theorem 
2,  and  the  group  K  is  invariant  under  Hf  in  Theorem  2'. 
Any  group  is  an  invariant  subgroup  of  itself  (§  28). 

A  group  which  contains  no  invariant  subgroups 
(except  itself)  is  called  a  simple  group. 

EXERCISES 

1.  If  G  is  a  simple  group,  none  of  the  numbers  k2,  /Ca,   .    .     of 
Exercise  5,  §  30,  can  be  unity. 

2.  Prove  that  a  group  of  order  5  is  simple. 

More  generally,  a  group  whose  order  is  a  prime  number  is 
simple.  The  least  composite  number  which  can  be  the  order  of  a 
simple  group  is  60  (§  48). 

3.  The  operators  of  a  conjugate  set  of  a  group  G  generate  an 
invariant  subgroup  of  G. 

(Observe  that  the  operators  of  a  group  G'  generated  by  A,  B, 
C,  .  .  are  of  the  form 

T=    .    .    .  ApB9Cr  .    .  A^'C*  .    .    . 

The  condition  that  G'  be  transformed  into  itself  by  an  operator 
S  of  G  is  that  S~1TS  belongs  to  G'.  Now,  if  S~1AS,  S~1BS,  .  . 
are  denoted  by  Ai,  Bi,  .  .  ,  we  have  (Exercise  3,  §  25): 

S~1TS=   .    . 


40  FINITE  COLLINEATION  GROUPS 

In  other  words,  if  Gr  is  generated  by  A,  B,  C,  .  .  ,  then  S-WS 
is  generated  by  Ai,  Bi,  Ci,  .  .  .  But,  these  last  operators  belong 
to  the  conjugate  set  in  question  if  A,  B,  C,  .  .  do  that  (§  29),  (c) . ) 

4.  Prove  that  any  two  invariant  operators  of  G  are  commutative, 
and  that  their  product  is  also  an  invariant  operator  of  G. 

Hence  prove  that  all  the  invariant  operators  of  G  form  an 
abelian  group  H  (§  34)  which  is  invariant  under  G.  Furthermore, 
any  subgroup  of  H  is  an  invariant  subgroup  of  G. 

5.  Show  that  a  similarity-transformation  belonging  to  a  linear 
group  G  is  invariant  under  G,  and  that  the  group  of  similarity- 
transformations  contained  in  G  is  an  invariant  subgroup  of  G. 

6.  Prove  that  the  transformations  of  determinant  unity  con 
tained  in  a  linear  group  G  is  a  self-conjugate  subgroup  of  G. 

ISOMORPHISM,    §§  32-33 

32.  It  is  sufficiently  evident  from  the  preceding 
development  that  the  theory  of  groups  of  operators 
depends  entirely  upon  the  scheme  according  to  which  the 
products  of  operators  are  tabulated  (the  "multiplication 
table"  of  the  group.)*  Two  groups  whose  multiplication 
tables  are  the  same  have  essentially  the  same  abstract 
properties,  differing  only  in  the  notation  and  possibly 
the  meaning  of  their  operators.  The  relationship  is 
expressed  in  the  following: 

DEFINITION.  Two  groups,  G  and  K,  are  said  to  be 
simply  isomorphic  when  their  distinct  operators  are  equal 
in  number  and  can  be  arranged  in  relative  order  such  as : 

G:  E,  S2)  $3,   •    .   ,  Sg>' 
K:  E,  T2,  Tz,   .    .   ,  Tg, 

so  that  their  products  all  correspond;  that  is,  if  SaSb  = 
Se,  then  TaTb  =  Tc. 

Example  1. — The  groups  of  order  4  in  Examples  1  and  2,  §  27, 
are  simply  isomorphic. 

*  It  will  be  seen  later  (§  47)  that  if  a  multiplication  table  is  arbitrarily 
constructed,  so  that  only  the  conditions  of  §§  25  and  27  are  complied  with, 
then  there  is  at  least  one  group  of  operators  (a  substitution  group)  whoso 
multiplication  table  is  the  one  given. 


GROUPS  OF  OPERATORS  41 

On  the  other  hand,  these  groups  are  not  isomorphic  with  the 
group  G=(E,  S2,  S3,  84)  whose  operations  consist  in  multiplying 
by  1,  i,  —I,  —i  respectively,  where  i  —v—\.  The  square  of  every 
operator  of  the  first  two  groups  is  the  identity;  but  this  is  not  the 
case  with  S2  or  S*  of  the  present  group. 

Example  2. — Two  conjugate  groups,  H  =  (E,  A,  B,  .  .  )  and 
V-1HV  =  (E,  V~1AV,  V-WV,  .  .  ),  are  simply  isomorphic. 

A  more  general  kind  of  isomorphism  may  sometimes 
be  established  between  two  groups,  K  of  order  k  and  G 
of  order  g  =  kh.  For  instance,  let  it  be  possible  to  arrange 
their  operators  in  the  following  manner: 

K:  G: 

Ti  =  E;  $11,  $12,   .    .   ,  $IA; 

(3)  TZ'J  $21,    $22,     •      •     ,    $2A/ 

Tkt  Ski,   SfcZ,    •     •     ,    Skh,' 

so  that  first,  to  each  operator  of  K  there  correspond  h 
operators  of  G;  and  second,  to  each  product  $aa$&/3  = 
$CT  there  corresponds  a  product  TaTb=Tc,  irrespective 
of  the  subscripts  a,  /?,  y.  In  such  a  case  we  say  that  G 
is  (h,  1)  isomorphic  (or  multiply  isomorphic)  with  K. 
Concerning  two  such  groups  we  have  the 

THEOREM  3.  The  h  operators  of  G  which  correspond 
to  the  identity  of  K,  namely  $n,  $12,  .  .  ,  $IA,  form  an 
invariant  subgroup  of  G. 

First,  to  prove  that  $n,  $12,  .  .  ,  Sih  form  a  group 
H,  consider  any  product  $^$15.  By  the  conditions  of 
isomorphism,  this  product  must  be  an  operator  in  G  of  the 
set  which  corresponds  to  the  operator  TiTi  =  E2  of  K; 
that  is,  to  E.  Hence,  there  rnust  be  some  subscript  c  such 

that  $ia$i6  =  $ic. 

Next,  to  prove  that  H  is  invariant  under  G,  we  must 
show  that  $~1$ia$  belongs  to  H,  where  $  is  any  operator 
of  G.  Let  $  correspond  to  Tb  in  K,  and  $~1$ia$  will 


42  FINITE  COLLINEATION  GROUPS 

correspond   to   E,   since    Tb~lETb  =  E.     It   follows   that 
S~lSiaS  is  to  be  found  in  the  first  line;  i.e.,  it  belongs  to  H. 

EXERCISE 

The  linear  group  (6),  §  7,  is  (2,  1)  isomorphic  with  the 
collineation  group  G=(E,A*)  given  in  the  Example,  §  10. 

A  linear  group  is  always  isomorphic  with  the  corresponding 
collin cation  group. 

33.  Factor  group.  The  group  K  of  order  k  =  g/h 
discussed  in  the  preceding  paragraph,  is  called  a  factor 
group  (or  quotient  group)  of  the  group  G  of  order  g,  and  we 
write  symbolically 

K  =  G/H, 

H  being  the  invariant  subgroup  of  G  of  order  h  which 
corresponds  to  the  identity  of  K. 

THEOREM  4.  Let  G  be  a  group  of  order  g,  containing  an 
invariant  subgroup  H  of  order  h.  Then  a  factor  group  K  = 
G/H  of  k  =  g/h  operators  can  be  constructed,  to  which  there 
fore  G  is  (h,  1)  isomorphic. 

To  prove  this  theorem,  we  arrange  the  g  operators 
of  G  in  k  lines  as  shown  in  table  (1),  §  28,  the  operators 
of  H  forming  the  first  line. 

Now,  the  h2  "products  obtained  by  taking  for  a  pre- 
factor  in  turn  each  of  the  h  operators  from  a  given  line 
(the  ath)  and  as  a  post-factor  in  turn  each  of  the  h  opera 
tors  from  the  same  or  another  given  line  (the  6th),  will  all 
fall  in  one  line  (say  the  cth;  namely  the  line  in  which  the 
product  VaVb  falls);  symbolically  (HVa)(HVb)=HVc. 
This  is  easily  seen  in  the  following  manner.  Let 
(H)  represent  the  phrase  "an  operator  of  H,"  and  we 
have  (H)  (H)  =  (H) ;  and,  since  H  is  invariant  under  G, 
Va(H)  =  (H)  Va.  Consequently, 

(H)Va(H)Vb  =  (H)(H)VaVb=(H)Vc,  if  VaVb  =  (H)Vc. 

The  reader  may  now  readily  prove  that  the  symbols 
H,  HV2,  .  .  ,  HVic,  looked  upon  as  k  distinct  operators, 


GROUPS  OF  OPERATORS  43 

obey  the  conditions  1°—  6°,  §  25,  as  well  as  the  additional 
condition  for  a  finite  group  (§  27).  For  instance,  H 
satisfies  the  condition  for  the  identity:  (H)(HVa)  = 
(HVa)  (H)  =  (H)  Va.  There  is  therefore  at  least  one  group 
of  k  operators  E,  T2,  .  .  ,  Tk  having  the  same  multiplica 
tion  table  as  these  symbols  (§47).  Hence  the  existence 
of  the  factor  group  K,  as  a  group  of  operators,  is  proved. 

EXERCISES 

1.  Let  G  be  a  linear  group  in  n  variables,  and  H  the  group  of 
similarity-transformations  contained  in  G.     The  collineation  group 
corresponding  to  G  is  a  factor  group  G/H. 

2.  Prove  that  if  the  factor  group  K  contains  a  subgroup  K' 
of  order  A;',  then  the  corresponding  hk'  operators  of  G  form  a  sub 
group  G'  of  G. 

3.  If  X'  of  the  preceding  exercise  is  invariant  under  K,  then 
G'  is  invariant  under  G.  ^ 

4.  Prove  that  the  order  of  an  operator  of  G  is  divisible  by  the 
order  of  the  corresponding  operator  of  K.     (Hint:    if  S  belongs 
to  G  and  Sn=E,  then  the  corresponding  operator  T  of  K  satisfies 
the  equation  Tn=E.) 

TWO   SPECIAL  TYPES   OF   GROUPS,    §§  34-35 

34.  An  abelian  group  is  a  group  G  whose  operators 
E,  Sz,   .    .   ,  Sg  are  mutually  commutative: 

(4) 


The  following  two  propositions  are  immediate  con 
sequences  of  (4)  : 

(a)  Every  operator  of  an  abelian  group  G,  as  well  as 
every  subgroup  of  G,  is  invariant  under  G. 

(b)  A  factor  group  of  an  abelian  group  is  again  an 
abelian  group. 

Example.  —  Two  invariant  operators,  A  and  B,  of  any  group 
G  are  commutative,  since  A~1BA=B.  Hence,  the  invariant 
operators  of  G  form  an  abelian  self  -con  jugate  subgroup  of  G  (Exer 
cise  4,  §31). 


44  FINITE  COLLINEATION  GROUPS 

THEOREM  5.  An  dbelian  group  G  of  order  g  contains 
always  a  subgroup  G'  whose  order  g'  is  any  given  factor  of  g. 

In  the  proof  of  this  theorem  we  shall  adopt  the  process 
of  complete  induction.  Accordingly,  we  assume  the 
theorem  true  for  any  group  K  whose  order  k  is  less  than 
the  given  number  g,  the  order  of  the  given  group  G  for 
which  the  theorem  is  to  be  proved. 

Let  S  be  any  operator  of  G.  Assuming  for  the  moment 
that  its  order  h  is  a  factor  of  g',  we  construct  the  group  H 
of  order  h,  consisting  of  S  and  its  powers,  and  thereupon 
the  factor  group  K  =  G/H.  This  factor  group  being 
abelian  and  having  its  order  g/h  less  than  g,  it  contains 
by  assumption  a  subgroup  K'  of  order  g'/h,  a  factor  of 
g/h.  To  K'  will  now  correspond  a  subgroup  of  G  of  order 
g'  (Exercise  2,  §  33). 

/  The  proof  of  Theorem  5  thus  depends  on  our  finding 
an  operator  S  in  G  whose  order  is  a  factor  of  gr.  Now, 
the  order  n  of  any  operator  T  chosen  at  random  will 
either  be  prime  to  g',  or  will  contain  a  factor  h  which 
divides  g'.  In  the  latter  case  the  operator  Tn/h  may  be 
substituted  for  S  of  the  proof,  since  it  is  of  order  h  (§26, 
(6)). — In  the  first  case  let  N  be  the  group  formed  by  T 
and  its  powers.  The  factor  group  G/N  is  of  order  g/n, 
a  number  which  in  the  present  instance  is  divisible  by  gr. 
This  factor  group  therefore  contains  a  subgroup  of  order 
gf,  and  this  again  an  operator  whose  order  is  a  factor  of 
g',  say  h  (or  g'  itself).  The  order  of  a  corresponding 
operator  of  G  is  a  multiple  of  h  (Exercise  4,  §  33),  and  a 
power  of  this  operator  can  therefore  be  taken  for  S,  as 
shown  above. 

The  abelian  groups  of  order  pq  are  simply  isomorphic,  p  and  q 
representing  two  distinct  prime  numbers.  To  take  a  special  case, 
let  p  =  2  and  q  =  3.  An  abelian  group  of  order  6  contains  an  opera 
tor,  A,  of  order  2  and  one,  B,  of  order  3.  The  operator  AB  is  of 
order  6  and  will  generate  the  group  in  question. 


GROUPS  OF  OPERATORS  45 

If  p  =  q  the  matter  is  different:  there  are  two  distinct  types 
of  abelian  groups  of  order  pz.  One  type  is  generated  by  an  operator 
of  order  p2;  and  the  other  by  two  operators,  both  of  order  p.  If 
p  =  2  the  two  types  are  as  follows  (S  is  of  order  4,  A  and  B  both  of 
order  2)  : 

E,  S,  S2,  S3; 

E,A,B,  AB. 

35.  Groups  whose  orders  are  powers  of  a  prime  num 
ber.  The  study  of  these  groups  is  of  the  greatest  impor 
tance  for  the  theory  of  groups  in  general  as  well  as  for 
linear  groups,  particularly  in  view  of  Theorem  7.  We 
shall  here  prove  the  following  : 

THEOREM  6.  A  group  G  whose  order  is  pa,  p  being  a 
prime  number,  contains  p  or  more  invariant  operators, 
forming  an  invariant  subgroup  H.  If  G  is  not  abelian,  the 
abelian  subgroup  H  is  contained  in  a  larger  abelian  subgroup 
GI  which  is  invariant  under  G,  though  the  operators  of  G\  are 
not  all  separately  invariant  under  G. 

1°.  To  prove  the  existence  of  H,  we  construct  all  the 
sets  of  conjugate  operators  of  G  (§  29).  If  the  number 
of  sets  that  contain  just  one  operator  each  (invariant 
operators)  is  h',  and  if  the  remaining  sets  contain  respec 
tively  </2,  .  .  ,  gn  operators,  we  have  (cf.  Exercise  5, 
§30): 

(5)  p«  =  h'+gz+   .    .    .   +gn. 


The  numbers  gfe,  -  .  ,  gn  are  all  powers  of  p,  being  factors 
of  pa  and  greater  than  unity.  It  follows  that  h'  must  be 
divisible  by  p.  Hence,  G  contains  at  least  p  invariant 
operators,  and  all  such  operators  form  an  abelian  self- 
conjugate  subgroup  (Example,  §  34),  say  of  order  pb 
(Theorem  1,  §  28).  If  G  is  abelian,  pb  =  pa. 

2°.  To  prove  the  existence  of  G\,  we  now  construct 
the  factor  group  K  =  G/H  of  order  pa~b.  Treating  this 
in  the  same  way  as  G  above,  we  find  an  abelian  subgroup 


46  FINITE  COLLINEATION  GROUPS 

KI  consisting  of  invariant  operators  of  K.  Let  T  be 
any  one  of  these  except  the  identity,  and  let  pc  be  the 
order  of  that  subgroup  of  K\  formed  by  T  and  its  powers. 
This  subgroup  is  invariant  under  K,  and  the  correspond 
ing  subgroup  of  G  of  order  pb+c  is  invariant  under  G 
(Exercise  3,  §  33). 

We  have  still  to  prove  that  this  subgroup  Gi  is  abelian. 
By  referring  to  the  table  (3),  §  32,  we  see  that  G\  is  gen 
erated  by  H  and  one  of  those  operators,  V,  of  G  corre 
sponding  to  T  of  K.  Now,  all  the  operators  of  H  are 
commutative  with  each  other  and  with  V,  and  the  latter 
is  commutative  with  any  power  of  itself.  The  operators 
of  Gi  are  therefore  mutually  commutative. 

EXERCISES 

1.  Prove  that  a  group  of  order  p2  is  abelian,  and  is  therefore 
isomorphic  with  one  of  the  types  given  in§  34. 

2.  The  process  above  may  be  extended  as  follows:    to  K\ 
corresponds  an  invariant  subgroup  H\  of  G;  the  factor  group  G/H\ 
contains  a  subgroup  consisting  of  invariant  operators,  and  to  this 
subgroup  will  correspond  an  invariant  subgroup  #2  of  G,  etc.     In 
this  manner  we  shall  obtain  a  series  of  subgroups  of  G: 

H,  HI,  HZ,  HZ,  ..,(?. 

Prove  that  each  is  an  invariant  subgroup  of  all  that  follow  it,  and 
that  the  factor  groups  Hi/H,  H^/Hi,  .  .  ,  are  abelian. 

3.  Prove  that  a  group  of  order  po  contains  a  subgroup  of  order 
pt>,  if  b<a. 

SYLOW'S   THEOREM,    §§  36-39 

36.  In  the  case  of  a  non-abelian  group  the  sweeping 
statement  of  Theorem  5,  §  34,  is  not  generally  true.  But 
we  can  always  predict  a  certain  class  of  subgroups  called 
Sylow  subgroups,  as  stated  in  the  following: 

THEOREM  7.  (a)  Let  G  be  a  group  of  order  g,  and  let 
p  be  any  prime  factor  of  g.  If  pn  is  the  highest  power  of  p 


GROUPS  OF  OPERATORS  47  » 

that  divides  g,  then  there  is  in  G  at  least  one  subgroup  oj 
order  pa.     Denote  this  subgroup  by  P. 

(b)  If  G  contains  more  than  one  subgroup  P,  then  all 
such  subgroups  are  conjugate  under  G,  and  their  number  is 
of  the  form  l+pk. 

Proof  of  (a).  —  We  adopt  the  process  of  complete 
induction,  assuming  that  any  group  whose  order  is  less 
than  the  given  number  g  and  is  divisible  by  a  power  of 
p,  say  pb,  but  by  no  higher  power  of  p,  contains  a  subgroup 
of  order  pb.  Two  cases  arise:  1°,  G  contains  an  invariant 
operator  of  order  p;  2°,  G  contains  no  such  invariant 
operator. 

1°.  Let  S  be  an  invariant  operator  of  order  p,  and  let 
H  be  the  subgroup  formed  by  S  and  its  powers.  Consider 
the  factor  group  G/H-  Its  order  is  g/p,  a  multiple  of 
pa~l.  By  assumption,  it  has  a  subgroup  of  order  pa-1. 
To  this  subgroup  corresponds  a  subgroup  of  G  of  order 
pa-ip  =  pa  (Exercise  2,  §  33). 

2°.  Here  there  may  or  may  not  be  invariant  operators 
in  addition  to  E  in  G  whose  orders  are  prime  to  p.  All 
such  operators  form  an  abelian  group  H'  (Exercise  4,  §  31). 
The  order,  h',  of  this  group  is  prime  to  p.  For,  other 
wise  H'  would  contain  a  subgroup  of  order  p  (Theorem  5, 
§  34),  and  the  arguments  of  1°  would  be  valid. 

We  now  construct  all  the  complete  sets  of  conjugate 
operators  in  G,  and  obtain  an  equation  corresponding  to 
(5),  §35: 

.    +gn. 


Evidently,  the  numbers  </2,  .  .  ,  gn  cannot  all  be  mul 
tiples  of  p,  since  h'  is  not,  whereas  g  is  a  multiple  of  pa. 
Hence,  at  least  one  of  the  numbers  02,  .  .  ,  gn,  say  02, 
is  prime  to  p.  If  therefore  S  is  an  operator  in  this  set 
of  conjugates,  we  conclude  by  the  aid  of  Theorem  2,  §  29, 
that  G  contains  a  subgroup  H  of  order  h  =  g/gz,  a  number 


48  FINITE  COLLINEATION  GROUPS 

which  is  less  than  g  and  is  divisible  by  pa.  By  assump 
tion,  this  group  contains  a  subgroup  of  order  pa  which  is 
a  subgroup  of  G,  since  H  is  a  subgroup  of  G. 

37.  In  the  proof  of  (6)  the  following  propositions  are 
needed : 

1°.  An  operator  T  of  G  of  order  pa  which  transforms 
a  Sylow  subgroup  P  of  G  into  itself  (T~1PT  =  P)  belongs 
to  P. 

2°.  If  two  different  Sylow  subgroups,  PI  and  P2,  have 
in  common  a  subgroup  of  order  pb,  b<a  (cf.  Exercise  6, 
§  27),  then  one  of  them  is  transformed  into  just  pa~b  dis 
tinct  groups  of  order  pa  by  the  operators  of  the  other. 

To  prove  1°,  let  ra  be  the  least  positive  number  such 
that  Tm  belongs  to  P;  in  any  event,  m^p".  Then  we 
can  show  that  m  is  a  factor  of  pa;  for  if  it  is  not,  let  q  be  the 
quotient  and  r  the  remainder  when  pa  is  divided  by  m, 
so  that  pa  —  mq  =  r;  r<m.  Then,  since  Tpa  and  Tmq  are 
both  contained  in  P,  it  follows  that  Tr  is  also  contained 
in  P,  contrary  to  the  assumption  as  to  Tm. 

Therefore  m  is  one  of  the  numbers  1,  p,  p2,  .  .  . 
Assuming  that  T  does  not  belong  to  P,  let  m  =  p^.  The 
group  Q  generated  by  T  and  the  operators  of  P  will  now 
be  of  order  mpa  =  pP+a,  since  it  consists  of  all  the  operators 

P,  PTt  PT2,   .    .   ,  PTm~\ 

(That  these  operators  form  a  group  of  order  mp<*  is  seen  as 
follows:  If  we  indicate  the  phrase  "an  operator  of  P"  by  the  symbol 
(P),  we  have  T8(P)  =  (P)TS,  since  T~1PT  =  P,  and  therefore 
((P)T8)((P)Tt)  =  (P}(P)TsTt  =  (P)Ts+i,  so  that  the  products  of 
operators  of  the  set  Q  are  all  contained  in  the  set.  Again,  the  mpa 
operators  are  all  distinct.  For  the  pa  operators  PTS  are  all  distinct; 
and  the  equation  (P)T3  =  (P)Tl  cannot  be  true  unless  (P)  =  Tt~8; 
that  is,  unless  s  =  l.) 

But  G  cannot  contain  a  subgroup  Q  of  order  p^+° 
(Theorem  1,  §  28).  Hence,  T  must  belong  to  P. 


GROUPS  OF  OPERATORS  49 

To  prove  2°,  let  H  be  the  subgroup  of  order  pb  common 
to  the  two  groups  PI  and  P2.  By  1°,  no  operator  of  PI 
except  those  of  H  transform  P2  into  itself.  We  now  adopt 
the  process  employed  in  the  proof  of  Theorem  2,  §  29  : 
we  arrange  the  operators  of  PI  in  the  form  of  the  table 
(1),  §  28,  the  first  line  consisting  of  the  operators  of  H. 
To  each  line  of  the  table  there  will  then  correspond  a  dis 
tinct  group  conjugate  to  P2,  making  in  all  pa/pb  =  pa~b 
distinct  groups. 

38.  Proof  of  (b).  —  Consider  a  set  of  conjugate  Sylow 
subgroups  of  order  pa: 

(6)  Pi,  P,,   .    .   ,  P.. 

By  2°,  §  37,  the  operators  of  PI  will  transform  P2  into 
pa~b  distinct  groups.  If  the  set  contains  other  groups 
in  addition  to  these  pa~b  and  PI,  then  the  operators  of 
this  group  (Pi)  will  transform  one  of  them  into  say 
pa-c  groups  distinct  from  those  already  counted.  Pro 
ceeding  thus,  we  find 


since  pa~b,  pa~c,   .    .  are  all  multiples  of  p. 

Now,  if  G  contained  another  set  Pn+i,  -  .  ,  Pm  of 
conjugate  Sylow  subgroups,  their  number  would  likewise 
be  of  the  form  l-\-pk'.  On  the  other  hand,  PI  is  not  a 
member  of  this  new  set.  We  can  therefore  show  that  the 
number  of  groups  in  the  set  is  a  multiple  of  p}  say  pk", 
in  the  same  manner  as  the  number  of  groups  P2,  .  .  ,  Pn 
were  shown  to  be  a  multiple  of  p.  But,  the  equation 
l+pk'  =  pk"  is  impossible.  It  follows  that  (6)  is  the  only 
set  of  Sylow  subgroups  contained  in  G,  and  (b)  is  fully 

proved. 

EXERCISES 

1.  Prove  that  a  group  G  whose  order  is  divisible  by  pb  con 
tains  a  subgroup  of  order  pb.  (Cf.  Exercise  3,  §  35.) 


50  FINITE  COLLINEATION  GROUPS 

2.  If  the  largest  subgroup  H  that  PI  can  have  in  common  with 
another  Sylow  subgroup  is  of  order  pb,  prove  that  the  number  of 
Sylow  subgroups  in  G  of  order  pa  is  of  the  form  1  -{-pa~bk. 

3.  Show  that  a  group  of  order  40  contains  a  single  Sylow  sub 
group  of  order  5,  which  is  therefore  an  invariant  subgroup. 

4.  Prove  that  a  group  of  order  pq  is  abelian,  if  p  and  q  are  prime 
numbers  such  that  p  —  1  is  not  divisible  by  q,  nor  q  —  1  divisible  by  p. 

39.  We   conclude   by    proving   the   following   useful 
theorem  : 

THEOREM  8.  A  subgroup  of  G  of  order  pb  is  always 
contained  in  at  least  one  Sylow  subgroup  of  order  pa. 

Let  K  be  a  subgroup  of  order  pb,  and  assume  it  is  not 
contained  completely  in  any  one  of  the  groups  PI,  P2, 
.  .  ,  Pn,  §  38.  Then,  arranging  these  in  sets  of  conju 
gates  with  respect  to  the  operators  of  K,  we  find  that  the 
number  of  groups  in  the  set  is  a  multiple  of  p,  as  would 
be  the  case  with  the  number  of  groups  in  the  tentative 
set  Pn+i)  •  •  ,  Pm,  §  38.  But  this  is  impossible  by 
Theorem  7,  (6). 

B.    SUBSTITUTION  GROUPS 

40.  Definitions,     (a)  A  substitution  (or  permutation) 
S  on  a  given  set  of  letters   0-1,  02,  a3,   .    .   ,  an  is  the 
operation  of  replacing  these  letters  respectively  by  ap, 
aq,  ar,   .    .   ,  aw,  with  the  understanding  that  the  sub 
scripts  p,  q,  r,  .    .   ,  w  are  the  subscripts  1,  2,  3,   .    .   ,  n 
over  again  in  the  same  or  a  different  order.*     We  shall 
temporarily  indicate  such  a  substitution  by  the  symbol 


(7) 

aq  ar 

*  The  letters  are  generally  involved  in  one  or  more  given  functions, 
as  a1z1+o2z,+a,z3+  .  .  +anxn.  This  function  would  be  transformed 
into  o-Zj  +aqx2  +arx3  +  .  .  +awxn  by  the  substitution  S  above. 

Tne  phrases  "replace  a  by  b,"  "put  b  in  place  of  a,"  "change  a  into 
6"  are  to  be  regarded  as  synonymous. 


SUBSTITUTION  GROUPS  51 

The  columns  may  evidently  be  changed  about  in  any 
manner  so  long  as  ap  falls  below  a\,  aq  below  02,  etc. ;  and 
two  substitutions  are  equal  if  their  columns  are  equal  aside 
from  the  order  in  which  they  are  written.  For  instance, 

/Oi    02    «3\  _  /#2    «1    «3\     ,     /«!    «2    # 

W  a3  aj  ~  \as  «2  aj     \Oi  o3  ( 
(b)  The  product  ST  of  two  substitutions 

£       /Ol     «2     «3     •      •     0»\  T=(ap    a9    ar 

\ap  aq  ar   .    .  o./'  \aa  a&  Oy 

is  the  substitution  resulting  from  carrying  out  the  two 
substitutions  successively,  first  S  and  then  T: 


With  this  rule  for  multiplication,  the  associative  law: 
S(TU)  =  (ST)U,  is  readily  found  to  be  true. 

(c)  The  identity  is  the  substitution  in  which  each  letter 
is  replaced  by  itself: 


«i  02 
ai  02 


.    .  on\ 
.    .  aj' 

(d)  The  inverse  of  the  substitution  S  is  the  substitution 

ar  .    .  aw\ 
o3    .    .  on/ 


02 

In  consequence  of  these  definitions  and  propositions,  the  totality 
of  substitutions  on  n  letters  ai,  az,  .  .  ,  an  form  a  class  O  of  operators 
(§25). 

41.  Permanent  notation.  A  simpler  notation  for  a 
substitution  S  than  that  employed  in  (7),  §  40,  may  be 
developed  as  follows.  Writing  down  the  letter  a\  we 
follow  with  the  letter  that  takes  its  place,  namely  ap;  then 


52  FINITE  COLLINEATION  GROUPS 

we  follow  with  the  letter  that  replaces  ap,  say  aa;  and 
so  on:  aiapaa  .  .  .  There  being  a  finite  number  of 
letters,  we  must  sooner  or  later  arrive  at  a  letter  that  has 
already  been  written.  The  first  one  to  be  thus  duplicated 
must  be  «i/  for,  if  it  were  some  other  letter,  say  ap,  the 
lower  line  apaq  .  .  aw  of  the  substitution  S  in  (7)  would 
plainly  have  to  contain  ap  at  least  twice,  whereas  the 
letters  of  this  line  are  the  n  different  letters  0102  .  .  an 
in  some  order,  once  each. 

Let  therefore  the  last  letter  to  be  written  down  before 
a\  reappears  be  am.    We  then  have  a  cycle 

(aiapaa  .    .    .  am). 

If  this  cycle  does  not  exhaust  the  n  letters  under  considera 
tion,  we  start  with  a  new  letter  and  proceed  as  above, 
forming  a  new  cycle,  which  may  be  written  immediately 
after  the  first;  and  so  on.  No  one  letter  will  appear  in  two 
different  cycles,  as  otherwise  such  a  letter  would  have  to 
appear  at  least  twice  in  the  lower  line  of  (7).  It  is  cus 
tomary  to  exclude  all  cycles  which  contain  just  one  letter, 
such  a  letter  remaining  unchanged  by  S.  Thus  the  sub 
stitution 

o  _  /01    02    03    04    05    06\ 

\ai  a3  a4  02  «6  05/ 
is  written  S=  (020304)  (050e). 

EXERCISES 

1.  Show    that    if    S-(0l<wwli).  and  T  -(******),  then 

\a2030405ai/  VaiasO^s^/ 


Show  also  that  in  the  permanent  notation,  £  =  (0102030405), 
T  =  (0203)  (o4a6),  and  that  ST  =  (oia3a5),  TS  =  (010204). 

2.  The  substitution  (020304)  (a5a6)  is  the  product  of  the  sub 
stitutions  (020304)  and  (ooOc).  Prove  that  a  substitution  consisting 
of  a  number  of  cycles  is  equal  to  the  product  of  the  substitutions 


SUBSTITUTION  GROUPS  53 

represented  by  the  individual  cycles,  and  that  these  factors  are 
commutative. 

3.  Show  that  the   inverse  of   the    cycle   (a\a^  .    .  a,n—iQn)  is 
the  cycle  (anan-i   .    .  a^i),  and  that  the  inverse  of  a  substitution 
consisting  of  a  number  of  cycles  is  the  substitution  consisting  of  the 
inverses  of  those  cycles. 

4.  Show  that  the  order  of  the  operator  consisting  of  a  single 
cycle  containing  n  letters  is  n.     Hence  prove  that  the  order  of  a 
substitution   composed   of  several   cycles   containing  respectively 
Wi,  n2,   .    .   letters,  is  the  least  common  multiple  of  the  numbers 
Wi,  ft-;,   .    .   .     (Illustration:   the  order  of  (a^scu)  (asfle)  is  6.) 

5.  The  order  of  ST  in  Exercise  1  above  is  3,  and  the  cycles  in 
S  and  T  contain  respectively  5  and  (2,  2)  letters.     Do  these  facts 
agree  with  the  statements  in  Exercise  4  ? 

42.  Even  and  odd  substitutions.  A  cycle  of  k  letters 
is  equal  to  the  product  oi  k—l  cycles  of  2  letters  each 
(called  transpositions)  : 


We  therefore  classify  substitutions  into  even  or  odd  sub 
stitutions  according  to  whether  they  are  equal  to  a  prod 
uct  of  an  even  or  an  odd  number  of  transpositions.  For 
instance,  the  substitution  (aiCfeOs)  (a4a5a6a7)  is  odd,  since 
it  can  be  written  as  the  product  (4i<fe)(4itife)(aiai)(aiai) 
(a4a?). 

This  classification  is  justified  since,  no  matter  in  which 
one  of  the  infinite  number  of  possible  ways  a  given  sub 
stitution  can  be  written  as  a  product  of  transpositions,  it 
is  either  always  even  or  always  odd.  To  prove  this  state 
ment,  consider  the  following  function: 

f=(ai  —  02)  (ai  —  a3)   .    .    (ai  —  ajfe  —  a3)   .    .    (02—  an)   .    . 
(an_i  —  an), 

*  Note  that  the  letter  o,  is  common  to  all  the  transpositions.  Hence, 
this  is  not  the  normal  notation  for  a  substitution  as  developed  in  §  41. 
The  laws  deduced  in  Exercises  3  and  4  are  therefore  not  applicable  to  the 
form  above. 


54  FINITE  COLLINEATION  GROUPS 

namely  the  product  of  all  the  possible  differences  of  the 
n  letters  involved.  This  function  changes  sign  when 
operated  upon  by  a  single  transposition,  and  hence  also 
when  operated  upon  by  an  odd  substitution,  but  remains 
unchanged  in  value  when  operated  upon  by  an  even  sub 
stitution. 

43.  Substitution  groups.  Symmetric  and  alternating 
groups.  A  set  of  g  different  substitutions  form  a  sub 
stitution  group  of  order  g  if  the  product  of  any  two 
substitutions  of  the  set,  whether  equal  or  not,  is  again  a 
substitution  of  the  set.  Cf.  §  27,  where  such  a  group  is 
given  in  Example  2.  In  our  present  notation  this  group 
is  written  as  follows: 

(8)  E,  (ab)(cd),  (ac)(bd),  (ad)(bc). 

All  the  n\  permutations  on  n  letters  evidently  form  a 
group,  called  the  symmetric  group  on  the  given  letters. 
Thus,  the  symmetric  group  on  3  letters  a,  6,  c  is  of  order 
6  and  is  composed  of  the  substitutions  E,  (abc),  (ac6), 
(06),  (ac),  (be). 

It  is  plain  that  the  totality  of  the  even  substitutions 
contained  in  the  symmetric  group  G  on  n  letters  form  by 
themselves  a  group,  called  the  alternating  group.  Its 
order  is  one-half  that  of  the  corresponding  symmetric 
group.  For,  let  there  be  p  even  and  q  odd  substitutions 
in  G.  The  p+q  products  obtained  by  multiplying  each 
of  these  substitutions  by  a  given  transposition  must  pro 
duce  the  same  p+q  substitutions  over  again  (Exercise  3, 
§  27)  with  this  difference,  that  we  now  have  p  odd  and 
q  even  substitutions.  Hence  p  =  q. 

Inside  a  given  symmetric  group,  the  conjugate  of  an 
even  substitution  is  again  an  even  substitution.  It 
follows  that  the  alternating  group  is  invariant  (§  31)  under 


SUBSTITUTION  GROUPS  55 

the    corresponding   symmetric    group.     The   alternating 
group  on  5  or  more  letters  is  always  a  simple  group  (§  49). 

EXERCISE 

Construct  the  symmetric  and  alternating  groups  on  4  letters 
a,  6,  c,  d.  Show  that  the  latter  contains  a  single  Sylow  subgroup 
of  order  4,  namely  the  group  (8),  and  that  the  former  contains  3 
Sylow  subgroups  of  order  8,  which  all  contain  the  group  (8). 

44.  Transitivity  and  intransitivity.     If  the  letters  of 
a  substitution  group  break  up  into  two  or  more  sets  hav 
ing  no  letters  in  common,  such  that  no  substitution  will 
replace  a  letter  of  one  set  by  a  letter  of  another,  then  the 
group  is  said  to  be  intransitive.     Otherwise  the  group  is 
transitive. 

The  transitive  groups  are  further  subdivided  into  primitive  and 
imprimitive  groups.  If  the  letters  break  up  into  two  or  more  sets 
of  such  a  nature  that  the  letters  of  any  one  set  are  either  all  replaced 
by  letters  of  the  same  set,  or  are  all  replaced  by  letters  of  another 
set,  then  the  group  is  imprimitive.  If  no  such  division  is  possible, 
the  group  is  primitive. 

Examples. — The  group  (8),  §  43,  is  transitive,  while  the  group 
E,  (ab)(cd),  is  intransitive.  The  two  sets  of  letters,  (a,  6)  and 
(c,  d)  are  called  systems  (or  sets)  of  intransitivity. 

The  transitive  group  (8),  §  43,  is  imprimitive.  Its  two  systems 
of  imprimitivity  may  be  selected  in  three  ways:  1°:  (a,  6),  (c,  d)\ 
2°:  (a,  c),  (6,  d);  3°:  (a,  d),  (6,c).  The  symmetric  group  on  three 
or  more  letters  is  primitive. 

45.  We  proceed   to   prove  the  following  important 
theorem  concerning  transitive  groups: 

THEOREM  9.  Let  G  be  a  transitive  substitution  group  of 
order  g  on  n  letters  a\,  02  .  .  ,  an.  Then, 

(a)  there  is  in  G  a  substitution  which  replaces  any  given 
letter,  say  a\,  by  any  other  given  letter; 

(b)  all  those  substitutions  in  G  which  leave  unchanged  a 
given  letter,  say  a\,  form  a  subgroup  of  order  g/n. 


56  FINITE  COLLINEATION  GROUPS 

Proof.  —  (a)  Let  us  assume  that  ai,  02,  .  .  ,  am 
(m<n)  are  the  letters  into  which  a\  is  changed  by  the 
various  substitutions  of  G.  Then  none  of  these  m  letters 
can  be  replaced  by  one  of  the  remaining  letters  am+i, 
.  .  ,  an;  or  vice  versa.  For,  let  V2  change  ai  into  0%,  and 
assume  that  there  is  a  substitution  T  which  changes  02 
into  Om+i',  then  the  substitution  V2T  would  change  a\ 
into  Om+if  contrary  to  what  was  stated  in  regard  to  the 
first  m  letters.  Again,  if  the  substitution  T\  changed 
one  of  the  last  n—  m  letters  into  one  of  the  first  m  letters, 
then  Ti~l  would  do  the  reverse.  But  this  has  just  been 
proved  impossible. 

Now,  a  transitive  group  cannot  contain  two  such 
sets  of  letters.  Accordingly,  G  must  contain  a  substitu 
tion  (Vz)  which  replaces  ai  by  02,  one  (F3)  which  replaces 
ai  by  as,  etc.,  and  finally  one  (Vn)  which  replaces  a\  by  an. 

(b)  Let  all  those  substitutions  which  leave  ai  un 
changed  constitute  a  set  H=(Si,  82,  .  .  ,  Sh),  then  H 
is  a  group.  For,  the  products  SaSb  all  belong  to  H. 

To  find  the  order  of  H,  we  arrange  the  hn  substitutions 
H,  HV2,  HV3,  .  .  ,  HVn  in  the  form  of  the  table  (1), 
§  28.  The  h  substitutions  in  any  one  line  are  evidently 
all  distinct  (since  SaVc  =  SbVc  would  necessitate  Sa  =  Sb)', 
moreover  the  substitutions  of  two  different  lines  are  dis 
tinct,  since  those  in  the  line  HVC  all  replace  ai  by  ac. 
Finally,  any  substitution  of  G  must  occur  in  our  table. 
For,  if  T  replaces  cti  by  ac,  then  TVc~l  belongs  to  H,  say 

c-1  =  Say  so  that  T  =  SaVc.     Hence,  g  =  hn,  or  h  =  g/n. 


C.    ON  THE  REPRESENTATION  OF  A  GROUP  OF  OPERATORS 
AS  A  SUBSTITUTION  GROUP 

46.  Theorem  10.  A  group  of  operators  G  which  con 
tains  a  conjugate  set  of  n  subgroups  (or  operators)  is  simply 
or  multiply  isomorphic  (§  32)  with  a  transitive  substitution 
group  on  n  letters. 


SUBSTITUTION  GROUPS  57 

1°.  If  the  n  subgroups  (or  operators)  are  designated 
ai,  02,  .  .  ,  a/i,  we  construct  a  substitution  group  K 
on  the  n  letters  «i,  02,  -  .  ,  an,  isomorphic  with  G,  in  the 
following  manner.  Let  S  be  any  given  operator  of  G,  and 
let  us  suppose  that  it  transforms  the  group  (operator)  «i 
into  ap,  02  into  aq,  etc.  : 


Then  the  subscripts  p,  q,  .  .  ,  w  are  all  different  and  must 
be  the  subscripts  1,  2,  .  ,  n  over  again  in  some  order. 
Accordingly,  we  can  construct  a  substitution  as  follows 
(using  the  notation  of  §  40) : 

Ol     02      .      .     < 

ap  aq    .    .  ( 

and  we  shall  associate  this  substitution  with  S. 

2°.  To  the  g  operators  Si,  82,  .  .  of  G  thus  correspond 
g  substitutions  Ti,  T2,  .  .  of  a  set  K;  and  to  prove  that 
these  substitutions  form  a  group  isomorphic  with  G  it  is 
sufficient  to  prove  that  if  SaSb  =  Sc,  then  TaTb  =  Tc 
(§§27,  32).  Now,  if  Sa  transforms  a  group  (operator)  a' 
into  the  group  (operator)  a",  and  Sb  transforms  o"  into 
a'",  then  Sc  transforms  a'  into  a'": 


Sc~la'Sc  = 


Again,  if  Ta  replaces  a'  by  a",  and  Tb  replaces  a"  by  a"', 
then  Tc=TaTb  replaces  a'  by  a"'.  The  isomorphism  is 
therefore  established. 

3°.  If  now  the  g  substitutions  obtained  above  are  all 
distinct,  the  isomorphism  is  simple.  If,  on  the  other  hand, 
we  find  that  several  operators  Sai,  $02,  .  .  ,  Sah  cor 
respond  to  a  single  substitution  Ta,  then  G  is  multiply 
isomorphic  with  the  group  K'  composed  of  the  totality  of 


58  FINITE  COLLINEATION  GROUPS 

distinct  substitutions  in  K.  For,  arranging  the  operators 
of  G  into  sets  in  such  a  manner  that  all  those  operators 
furnishing  the  same  substitution  are  thrown  into  one  set, 
we  find  that  these  sets  contain  the  same  number  of 
operators. 

Thus,  the  h  operators  SaiS~f  ,  Sa^"1 ,  .  .  ,  SahS~*  all  cor 
respond  to  the  identity  TaT~a1=E;  and  conversely,  if  Sn,  Sn,  .  .  , 
Sih  all  correspond  to  the  identity  E,  then  the  operators  SaiSu, 
SaiSu,  .  .  ,  SaiSih  all  correspond  to  one  and  the  same  substitution 
TaTi  =  Ta. 

4°.  Finally,  to  prove  that  the  substitution  group  thus 
constructed  is  transitive,  we  note  that  within  G  the 
groups  (operators)  a\t  0%,  •  •  an  form  a  complete  con 
jugate  set,  so  that  there  is  an  operator  in  G  which  trans 
forms  a\  into  any  given  group  (operator)  ap.  There  is 
therefore  within  K  a  substitution  which  replaces  the  letter 
a\  by  the  letter  ap. 

COROLLARY.  //  a  simple  group  (§31)  G  contains  a 
set  of  n  conjugate  subgroups  (or  operators),  then  we  can 
construct  a  transitive  substitution  group  on  n  letters  which 
is  simply  isomorphic  with  G. 

EXERCISES 

1.  The  linear  group  (7),  §  7,  contains  a  conjugate  set  of  two 
operators,  /?2  and  #4.     Construct  a  substitution  group  which  is 
(1,  4)  isomorphic  with  the  given  group. 

2.  The  collineation  group  of  Exercise  2,  §  12,  contains  a  set  of 
three  conjugate  operators,  namely  the  2d,  3d,  and  6th.     Show  that 
the  given  collineation  group  is  simply  isomorphic  with  the  symmetric 
group  on  three  letters. 

3.  Let  p,  q,  r,  s  denote  the  four  subgroups  of  order  3  contained 
in  the  symmetric  group  G  on  four  letters  a,  6,  c,  d.     Construct  the 
transitive  substitution  group  on  the  letters  p,  q,  r,  s,  isomorphic 
with  G. 

47.  Theorem  11.  A  group  G  of  order  g  can  be  repre 
sented  as  a  transitive  substitution  group  H  on  g  letters 


SUBSTITUTION  GROUPS  59 

(called  a  regular  substitution  group).  In  this  representa 
tion,  every  substitution  except  the  identity,  will  replace  every 
letter  by  a  different  letter. 

The  group  (8),  §  43,  is  regular. 

Let  the  operators  of  G,  as  well  as  the  letters  of  sub 
stitution,  be  denoted  by  TI  (or  E),  T2,  .  .  ,  Tg.  We  now 
associate  with  an  operator  Tp  of  G  the  substitution 


_/Tl       T2        .          .         Tg\ 

—  I  rrir    rpf  rrir  ]  > 

\±l       LI        .         .        lg/ 


O 

& 


where  Tfa  is  the  letter  which  in  G  represents  the  operator 
TaTp;  i.e.,  T'a  =  TaTp.  (Cf.  §40;  the  symbol  making 
up  the  right-hand  member  of  (9)  is  actually  a  "substitu 
tion,"  since  the  letters  in  the  lower  line  are  the  letters  in 
the  upper  line  written  in  some  order  (Exercise  3,  §  27).) 

In  no  case  is  T'a  =  Ta  unless  Tp=  the  identity  =7\; 
and  then  every  T'a  =  Tai  so  that  Si  becomes  the  identity. 
Furthermore,  no  two  substitutions  corresponding  to 
different  operators  can  be  equal  (from  TaTp  =  TbTp 
follows  Ta  =  Tb).  Hence,  if  we  find  that  SpSq  =  Sr  when 
ever  TpTq  =  Tr,  the  substitutions  Si,  .  .  ,  Sg  form  a 
group  H,  simply  isomorphic  with  G,  and  fulfilling  the 
conditions  of  the  theorem  if  it  is  transitive. 

Now,  since  the  series  T1}  T2,  .  .  ,  Tg  is  equivalent 
to  the  series  TiTk,  T2Tk,  .  .  ,  TgTk,  aside  from  the  order, 
the  substitution  Sq  may  equally  well  be  written 

O    _  I       *     k  •      '          ok        \ 

y       \  T"*  '7^  T^  '7^  '7^  '7^    /  ' 

V-^l^^^g      •      •      J-gJ-k^q/ 

which  we  shall  abbreviate  to  (TaTk,  TaTkTq).  We  then 
have 

SpSq  =  (Ta,  TaTp)(TaTp,  TaTpTq)  =  (Ta,  TaTpTq) 
=  (Ta,  TaTr)=Sr, 

and  the  isomorphism  is  proved. 


60  FINITE  COLLINEATION  GROUPS 

The  group  H  will  be  transitive  if  there  is  a  substitu 
tion  which  replaces  T\  by  any  given  letter  Tn.  Now,  the 
substitution  Sn=(Ta,  TaTn)  does  that.  The  theorem  is 
therefore  proved. 

EXERCISE 

Construct  the  regular  substitution  group  on  6  letters  x\t  x-2,  .  .  , 
x6,  isomorphic  with  the  symmetric  group  on  3  letters. 

D.    ON  SIMPLE  GROUPS 

48.  In  later  chapters  it  will  be  of  great  convenience 
for  us  to  use  a  number  of  known  results  about  simple 
groups.     Some  of  these  results  shall  merely  be  stated 
here  without  proof;   in  the  case  of  the  theorems  12  and 
13  the  proofs  are  outlined  for  the  benefit  of  advanced 
students.     The    detailed   analysis   would   be   somewhat 
lengthy  and  in  part  difficult. 

We  begin  by  enumerating  the  simple  groups  whose 
orders  are  not  greater  than  2000*  or  that  can  be  repre 
sented  as  substitution  groups  on  not  more  than  10 
letters :f 

(a)  the  alternating  groups  on  5,  6,  .  .  ,  10  letters 
(§49); 

(6)  certain  groups  of  orders  168,  504,  660,  1092. 

49.  Theorem   12.     The  alternating  substitution  group 
of  order  n\/2  on  n  letters  is  a  simple  group  when  ni=5. 

Outline  of  proof. — 1°.  If  a  group  on  n  letters  ai,  .  . ,  an 
contains  all  the  substitutions  of  the  form  (apaqar),  it  is  the 
alternating  group. 

*  Holder,  Mathematische  Annalen,  XL  (1892),  55;  Cole,  American 
Journal  of  Mathematics,  XIV  (1892),  378,  XV  (1893),  303;  Burnside, 
Proceedings  of  the  London  Mathematical  Society,  XXVI  (1895),  333;  Ling 
and  Miller,  American  Journal  of  Mathematics,  XXU  (1900),  13. 

Strictly  speaking,  a  group  whose  order  is  a  prime  number  is  a  simple 
group,  having  no  invariant  subgroups.  But  such  groups  will  not  be 
included  under  the  concept  "simple  groups"  in  succeeding  chapters. 

t  Jordan,  Comptes  Rendus,  LXXV  (1872),  1754. 


SUBSTITUTION  GROUPS  61 

2°.  A  possible  self-conjugate  subgroup  of  the  alternat 
ing  group  which  contains  a  substitution  S  must  contain 
the  substitution  U  =  S~1T-1ST,  where  T  is  any  substitu 
tion  in  the  alternating  group.  Now,  whatever  form  S 
may  have,  it  is  always  possible  to  find  such  a  substitution 
T  that  U  is  composed  of  the  single  cycle  (apaqar). 

3°.  All  the  conjugates  to  U  with  respect  to  the  sub 
stitutions  of  the  alternating  group  must  be  contained  in  the 
self-conjugate  subgroup.  But  these  conjugates  are  indeed 
all  the  possible  substitutions  composed  of  just  a  single 
cycle  of  three  letters  each;  and  therefore,  by  1°,  the  pro 
posed  self-conjugate  subgroup  is  the  alternating  group 
itself. 

50.  Theorem  13.*  The  alternating  group  on  n  letters 
is  simply  isomorphic  with  the  group  generated  by  the  n  —  2 
operators  F\,  .  .  ,  Fn-2  which  satisfy  the  relations: 


(FaFby  =  E      (a  =1,2,   .    .   ,  n-4;   6  =  a+2,  a+3, 
.   .   ,  n-2). 

Outline  of  proof.  —  1°.  Adopting  the  process  of  complete 
induction,  we  assume  that  FI,  .  .  ,  Fn_3  generate  a 
group  H  simply  isomorphic  with  the  alternating  group  on 
n  —  1  letters  (we  take  n  >  3  ;  f  or  n  =  3  the  theorem  is  self- 
evident).  Hence,  the  following  symbols: 

(10)  Rn,  Rn-i,  .    .  ,  Ri, 

where 
Rn-i  =  H,  Rk(k<n-l)  =  HFn.2Fn.3  .   .  Fk,  Rn  =  RiFlt 

represent  at  most  [(n—  l)!/2]n  =  n!/2  operators. 

*  E.  H.  Moore,  Proceedings  of  the  London  Mathematical  Society, 
XXVIII,  No.  596.  The  outline  given  above  is  of  the  proof  given  by  L.  E  . 
Dickson,  Linear  Groups  (Leipzig,  1901).  pp.  289-90. 


62  FINITE  COLLINEATION  GROUPS 

2°.  The  symbols  (10)  contain  all  the  operators 
generated  by  ^i,  .  .  ,  Fn_2.  For,  the  set  (10)  is  repro 
duced  if  we  multiply  on  the  right  by  any  one  of  these 
operators,  and  therefore  also  by  any  operator  generated 
by  them.  Accordingly,  the  group  G  generated  by  FI, 
.  .  ,  Fn-Z)  is  of  order  nl/2  at  most. 

3°.  Now,  the  substitutions  F(  =  (a^aa) ,  F^  =  (aiCfc)  (0304) , 
^3=(ai02)(a4a5),  •  •  >  Fn-z  =  (aide)  (an-\an)  satisfy  the  re 
lations  imposed  upon  the  corresponding  operators  of 
the  theorem.  It  follows  that  G  is  isomorphic  with 
the  substitution  group  G'  generated  by  F(,  .  .  ,  Fn-z. 
But  this  is  the  alternating  group  on  n  letters,  and  is  of 
order  w!/2. 

4°.  Finally,  the  order  of  G  being  at  most  w!/2  by  2°,  it 
follows  that  G  and  G'  are  simply  isomorphic. 


CHAPTER  III 
THE  LINEAR  GROUPS  IN  TWO  VARIABLES 

51.  General  remarks  on  linear  groups  and  colline- 
ation  groups. 

1°.  A  collineation  group  may  readily  be  exhibited  as  a 
linear  group,  and  vice  versa,  as  shown  in  §§9,  10,  12. 
It  is  therefore  necessary  to  discuss  only  one  of  these 
categories,  preferably  the  latter,  which  lends  itself  more 
readily  to  study.  Accordingly,  it  will  be  our  practice 
to  apply  the  descriptive  terms  of  chap,  ii,  (A),  to 
a  group  under  consideration,  with  reference  to  this  group 
as  written  in  linear  form.  For  instance,  we  shall  say  that 
the  group  (7),  §  7,  is  non-abelian,  though  the  corresponding 
collineation  group  (E,  AI,  BI,  B2)  is  abelian.  Moreover, 
certain  terms  have  reference  to  linear  groups  only, 
namely  (in)  transitivity  (§  14),  (im)primitivity,  and  mo 
nomial  form  (§  60). 

On  the  other  hand,  a  group  is  generally  (in  the  litera 
ture)  described  as  simple  if  the  corresponding  collineation 
group  is  simple;  that  is,  when  classifying  the  groups  in  a 
given  number  of  variables,  we  put  into  the  class  "  simple 
groups"  all  the  linear  groups  which  are  simple,  together 
with  those  whose  factor  groups  G/H  are  simple,  H  repre 
senting  the  group  of  similarity-transformations  contained 
in  G.  But  this  is  the  only  exception  to  the  practice 
agreed  upon. 

2°.  It  will,  furthermore,  be  our  practice  to  write  a 
linear  group  in  such  a  way  that  the  transformations  all 
have  a  determinant  unity.  Of  course,  if  G  (or  a  subgroup 
of  G)  is  intransitive,  then  a  constituent  of  this  group 
embracing  one  or  more  sets  of  intransitivity  may  not  be 

63 


64  FINITE  COLLINEATION  GROUPS 

subject  to  this  rule.  An  instance  is  furnished  by  an 
abelian  group  written  in  canonical  form:  (a,  a"1).  Here 
the  groups  in  one  variable,  say  x  =  axf,  do  not  have  unity 
for  the  value  of  their  determinants. 

3°.  It  is  often  convenient  to  write  the  order  of  a  group 
G  in  the  form  g<f>  (after  Jordan),  where  g  represents  the 
order  of  the  collineation  group  corresponding  to  G,  and 
<£  (not  specified)  the  order  of  the  group  of  similarity- 
transformations  contained  in  G.  For  example,  the 
order  of  the  group  (7),  §  7,  may  be  written  either 
8  or  4<£. 

We  may,  correspondingly,  write  the  order  of  a  linear 
transformation  S  in  the  form  </<£,  this  being  the  order  of 
the  group  generated  by  S.  Thus,  2<£  is  the  order  of  the 
transformation  (i,  —i),  where  i  =  V  —  1.  If  the  order 
of  the  transformation  S  =  (a,  /8)  is  g<j>,  that  of  the  trans 
formation  T=(l,  /3/a)  is  g. 

Obviously  we  write  a  linear  group  in  such  a  way  that  0  is  as 
small  as  possible,  under  the  condition  specified  in  2°.  In  the  case 
of  the  groups  in  two  variables,  <£  =  2,  except  in  one  instance,  since  a 
collineation  of  order  2  cannot  be  written  in  the  form  of  a  linear 
transformation  of  determinant  unity  unless  it  has  the  form  (i,  —i), 
where  i  =  V  —  1.  A  group  of  even  order  must  therefore  contain  the 
group  of  similarity-transformations  E  =  (l,  1),  E\  =  (  —  1,  —1). 
The  exception  mentioned  is  an  abelian  group  of  odd  order. 

4°.  Equivalence. — A  group  is  said  to  be  equivalent  to 
all  the  groups  which  flow  from  it  by  means  of  a  change  of 
variables.  If  the  groups  K\  and  K2  are  equivalent,  then 
there  is  a  linear  transformation  T  such  that  T~lKiT  =  K2 
(§  13).  Two  groups  equivalent  to  a  third  are  equivalent 
to  each  other  (cf.  §  30). 

52.  The  linear  groups  in  two  variables:  mode  of 
attack.  There  are  several  processes  available  for  the 
determination  of  the  different  non-equivalent  groups  in 


LINEAR  GROUPS  IN  TWO  VARIABLES  65 

two  variables.*  We  shall  here  employ  a  modified  form 
of  Klein's  original  process,  for  the  sake  of  its  historical 
and  geometrical  interest.  An  outline  of  this  process 
follows. 

1°.  Any  given  transformation  S  of  a  group  G  in  the 
variables  x\,  X2,  whose  Hermitian  invariant  is  I  =  XiXi-{- 
X2X2  can  be  written  as  a  product  S  =  818283,^  where  Si 
and  $3  have  the  canonical  form: 

81  =  (cos  u—i  sin  u}  cos  u-\-i  sin  u), 
S3=(cosw  —  i  sin  w,  cos  w+i  sin  w),  and 


Q  _  [    cos  v  sm  v~\ 
L  —  sin  v  cos  v J  * 


The  transformation  S  of  the  group  G  conjugate-imaginary 
to  G  is  similarly  equal  to  the  product  818283,  where 


81=  (cos  u+i  sin  u,   cos  u—i  sin  u), 
$3=  (cos  w-\-i  sin  w,  cos  w  —  i  sin  w?), 


(§53) 


We  shall  denote  by_H  the  intransitive  group  whose 
constituents  are  G  and  G;  that  is,  the  group  whose  vari 
ables  are  Xi,  fy,  x\y  x2.  In  these  variables  let  T,  TI,  T2,  T3 
denote  the  transformations  corresponding  to  S,  Si}  82,  83, 
and  we  have  T=TiT2T3. 

2°.  Now  let  the  three  real  functions  X  =  XiXi—  £2^2, 
Y  =  XiXz  +  xixi,  Z  =  (xiXz  —  £2X1)  V  —  1  denote  rectangular 
co-ordinates  in  space.  The  transformations  7\,  T2,  T3 

*  Klein,  Mathematische  Annalen,  IX  (1876),  183  flf.;  Vorlesungen  ilber 
das  Ikosaeder,  Leipzig,  1884,  pp.  116-20;  Gordan,  Mathematische  Annalen, 
XII  (1877),  23  ft.;  Jordan,  Journal  fiir  die  reine  und  angewandte  Mathe- 
matik,  LXXXIV  (1878),  93-112;  Atti  delta  Reale  Academia  di  Napoli, 
VIII  (1879);  Puchs,  Journal  fiir  die  reine,  etc.,  LXXXI,  LXXXV  (1876, 
1878),  97,  1  flf.;  Valentiner,  De  endelige  Transformations-Cruppers  Theori, 
Copenhagen,  1889,  pp.  100  ft*. 

t  The  transformations  Slt  S.2,  S3  are  not  separately  transformations 
of  G. 


66  FINITE  COLLINEATION  GROUPS 

are  then  shown  to  represent  real  rotations  around  the 
X— ,  Z  — ,  X— axes  respectively.  It  follows  by  a  well- 
known  theorem  that  T  is  a  real  rotation  around  a  certain 
axis  which  passes  through  the  origin.  There  results  a 
group  G'  of  rotations  in  space  which  is  isomorphic  with  G 
(§  54). 

3°.  Since  the  rotations  of  G'  leave  the  origin  fixed, 
they  must  transform  into  itself  a  sphere  2  whose  center 
is  the  origin.  If  now  R  be  an  axis  of  rotation  and  if 
PI  be  one  of  the  points  where  R  cuts  2,  then  PI,  and  all 
the  points  Pz,  .  .  ,  Pt  into  which  PI  is  transformed  by 
the  rotations  of  G'  will  be  the  vertices  of  a  regular  poly 
hedron  (including  the  limiting  cases  where  there  is  a  single 
axis  of  rotation  or  where  the  polyhedron  becomes  a  flat 
polygon)  (§55). 

4°.  The  groups  of  rotations  (G')  may  now  be  con 
structed  to  correspond  to  the  regular  polyhedra.  We 
find  five  types,  and  correspondingly  five  non-equivalent 
linear  groups  (G)  (§§  56-58). 

53.  Proof  of  1°. — The  transformation  S  has  the  unitary 
form  and  its  determinant  is  unity.  We  may  therefore 
write  (cf.  Exercise  1,  §  20): 


_T     a    6] 
1-6    aj' 


aa-\-bb=l. 


Let  p  and  q  represent  the  positive  square  roots  of  the 
real  positive  numbers  aa  and  bb  respectively,  and  we  have 
p2-\-qz=l.  Accordingly,  we  can  write  p  =  cos  v,  q  =  sin  v. 
Moreover,  \a/p\  =  1  and  \b/q\  =  1,  so  that  we  may  write 

a/p  =  cos  h— i  sin  h,          b/q  =  cos  k  —  i  sin  k; 
a/p  =  cos  h+i  sin  h,          b/q  =  cos  k+i  sin  k. 

If  we  finally  put  h  =  u+wt  k  =  u  —  w,  we  obtain  by  direct 
multiplication 

=  S. 


LINEAR  GROUPS  IN  TWO  VARIABLES  67 

54.  Proof  of  2°.—  The  functions  X,  Y,  Z  will  be 
transformed  by  7\,  T2,  T3,  defined  in  1°,  into  linear  func 
tions  of  themselves.  In  fact,  we  may  readily  prove  that 


(Z)Ti=  Y  sin  2u+Z  cos  2w, 

with  similar  results  for  T2  and  TV 

We  can  exhibit  these  results  in  a  different  form. 
Looking  upon  TI,  Tz,  T3  as  linear  transformations  in  the 
variables  X,  Y,  Z,  they  will  appear  as  follows  : 

TV  X  =  X',  Y=Yf  cos  2u-Z'  sin  2w, 

Z=Y'  sin  2u+Z'  cos  2w; 
TV  X  =  X'  cos2v+Y'sm2v,    Y=  -X'  sin  2v+F'  cos  2v, 


T3:  X  =  X',  Y=Y'  cos  2w-Z'  sin  2ti>, 

Z=Yf  sm2w+Zf  cos2w. 

If  we  interpret  X,  Y,  Z  as  rectangular  co-ordinates  in 
ordinary  space,  we  recognize  here  three  real  rotations 
around  the  X—,  Z  —  ,  X—  axes  respectively.  These 
rotations,  performed  successively,  are  equivalent  to  a 
single  rotation  T. 

To  the  different  transformations  (S)  of  G  will  in  this 
manner  correspond  rotations  (T7)  of  a  group  G',  isomorphic 
with  G.  The  isomorphism  is  (1,  2)  in  case  G  contains  the 
transformation  Ei=(—l,  —  1),  since  both  this  transforma 
tion  and  the  identity  (E=(l,  1)),  and  no  others,  give 
rise  to  the  single  rotation  E'=(l,  1,  1),  and  vice 
versa. 

55.  Proof  of  5°.  —  Consider  the  sphere  2  together  with 
all  of  the  axes  of  rotations  of  G'.  A  rotation  B  of  G' 
will  permute  these  axes  among  themselves;  in  fact,  the 
axis  of  a  rotation  A  is  by  B  transformed  into  the  axis  of 
the  rotation  B~1AB.  Accordingly,  if  PI  is  the  extremity 


68  FINITE  COLLINEATION  GROUPS 

of  an  axis  of  period  m  (that  is,  an  axis  whose  corre 
sponding  angles  of  rotation  are  the  different  multiples 
of  360°/w),  then  the  points  PI,  P2,  .  .  ,  Pt  into  which 
PI  is  transformed  by  the  various  rotations  of  G'  will  be 
extremities  of  axes  of  period  m,  and  the  distribution  of 
these  points  about  any  one  of  them  is  similar  to  the  distri 
bution  about  any  other. 

Now,  if  t>  1,  let  arcs  of  great  circles  be  drawn  connect 
ing  PI  with  all  the  points  P2,  .  .  ,  P*,  and  let  the  shortest 
arc  be  of  length  L.  The  number  of  arcs  of  this  length 
radiating  from  PI  is  m  or  a  multiple  of  m,  since  always  m 
of  the  arcs  are  interchanged  by  rotations  about  the  axis 
through  PI.  However,  there  cannot  be  more  than  5  such 
arcs,  unless  L  =  180°.  For,  if  there  were  6  or  more,  a 
pair  of  them,  say  Cs,  Cr,  would  make  an  angle  0=i600 
with  each  other  at  PI;  and,  this  being  the  case,  the  arc 
L'  connecting  Ps  and  Pr  (the  points  of  P2,  .  .  ,  Pt 
located  on  Ca  and  Cr)  would  have  a  length  <L.  For,  by 
trigonometry, 

cos  L'  =  cos2  L+sin2  L  cos  fli^cos2  L-\-\  sin2  L 
=  |  (l-f-cos2L)>cosL; 

and,  since  0<Z/<90°,  it  follows  that  L'<L.  But  this 
is  contrary  to  hypotheses,  since  the  lengths  of  the  arcs  ra 
diating  from  P8  are  equal  to  the  lengths  of  the  arcs 
radiating  from  PI.  Similarly  we  may  prove  that  an  arc 
of  length  L  joining  two  of  the  points  PI  .  .  P*  cannot 
intersect  another  arc  of  the  same  nature. 

Let  m>2.  Then  it  follows  that  there  are  just  m 
arcs  of  length  L  radiating  from  PI,  each  making  an  angle 
of  360°/ra  with  its  adjacent  arcs.  The  same  is  true  for 
each  of  the  points  P2,  .  .  ,  Pt,  and  we  see  that  the  sphere 
will  be  divided  by  all  the  arcs  of  length  L,  joining  the 
various  points  PI  .  .  Pt  which  can  be  reached  from  one 
of  them  by  passing  along  such  arcs,  into  a  number  of 


LINEAR  GROUPS  IN  TWO  VARIABLES  69 

equal  and  regular  polygons.     Accordingly,  these  points 
are  the  vertices  of  a  regular  polyhedron  inscribed  in  2. 

The  diameters  of  S  passing  through  the  middle  points  of  the 
arcs  L  and  through  the  middle  points  of  the  regular  polygons  will 
either  coincide  with  the  axes  already  obtained  or  will  be  additional 
axes  of  rotations  of  (?'. 

Next,  let  there  be  no  axis  of  period  greater  than  2. 
If  there  are  two  axes  of  period  2,  say  AI  and  A2,  cutting 
each  other  under  an  angle  a  which  may  be  assumed  ^90°, 
then  a  rotation  of  180°  around  AI  followed  by  a  rotation  of 
180°  around  A2  is  equivalent  to  a  rotation  of  2a  around  an 
axis  perpendicular  to  the  plane  of  AI  and  A2.  Hence 
2a  is  a  multiple  of  180°;  i.e.,  a  =  90°.  It  follows  that  we 
have  just  one  axis  of  period  2,  or  just  three  such  which 
are  mutually  perpendicular. 

THE  GROUPS  OF  THE  REGULAR  POLYHEDRA,  §§  56-58 

56  (4°) .  Limiting  cases.  In  order  to  tabulate  the  linear 
groups  (G)  we  may  proceed  as  follows.  From  the  geo 
metrical  data  given  the  analytical  equivalents  of  the  rota 
tions  of  G'  can  be  calculated,  and  then  we  can  reverse 
the  processes  of  1°  and  2°.  This  work  may  be  facilitated 
by  placing  any  given  configuration  arrived  at  in  3°  in  any 
convenient  position  with  reference  to  the  axes  of  co 
ordinates  X,  Y,  Z,  since  such  a  shifting  of  the  figure  is 
equivalent  to  a  change  of  variables  (x\,  £2)  in  the  respective 
group  G. 

Beginning  then  with  the  simplest  case  where  there  is 
a  single  axis  of  rotation,  we  let  this  be  the  X— axis.  Then 
sin  2i>  =  0  and  cos2y=l.  Hence  S  has  the  canonical 
form  (a,  a);  aa—  1. 

(A)  G':  a  single  axis  of  period  g; 

G:    an  abelian  group  (intransitive)  of  order  g: 

Sm=(*m,a-m);  m=l,2,    .    .   ,g;  a°  =  l. 


70  FINITE  COLLINEATION  GROUPS 

We  next  have  a  group  G'  containing  the  axis  of  period 
g  in  (A),  in  addition  to  g  axes  of  period  2  lying  in  the  plane 
X  =  0.  Let  one  of  the  latter  be  the  Z  —  axis ;  we  here  have 
cos2z;=— 1,  cos  2(u— w)  =  l,  and  the  corresponding 
transformation  of  G  is  found  to  be 


w= 


(B)  Dihedral  group. 

G'  :  one  axis  of  period  g  and  g  axes  of  period  2  ; 
G:    an  imprimitive  group  of  order  20<£  consisting 
of  the  transformations 


m-1,  2,   .    .  ,  g;  a*  =  l. 

57.  The    tetrahedron    and    octahedron.    We    now 

examine  the  five  ordinary  regular  solids.  Of  these,  the 
hexahedron  and  octahedron  furnish  the  same  set  of  axes 
of  rotation,  as  do  also  the  dodecahedron  and  icosahedron. 
We  therefore  have  only  three  cases  to  consider  :  the  tetra 
hedron,  octahedron,  and  icosahedron. 

In  the  case  of  the  tetrahedron  we  have  four  vertices 
and  correspondingly  four  axes  of  rotation  of  period  3; 
besides,  three  axes  of  period  2,  each  passing  through  the 
middle  points  of  a  pair  of  opposite  edges.  The  latter  are 
mutually  perpendicular  and  may  be  taken  as  the  X—, 
Y  —  ,  and  Z  —  axes.  The  corresponding  transformations 
of  G  are  then  as  follows: 


either  directly  or  after  multiplication  by  E\=(—  1,  —1). 
If  the  vertices  are  named  a,  b,  c,  d,  the  three  rotations 


LINEAR  GROUPS  IN  TWO  VARIABLES  71 

permute  them  among  themselves  according  to  the  sub 
stitutions 

(ab)(cd),  (ad)(bc),  (ac)(bd). 

The  remaining  rotations  permute  the  vertices  three 
at  a  time  cyclically,  as  (abc),  .  .  .  The  corresponding 
transformations  of  G  may  be  determined  analytically  from 
the  conditions  that  they  are  each  of  order  3  and  transform 
the  collineations  corresponding  to  Wi,  Wz,  Wz  cyclically. 
Certain  ambiguities  arise  from  the  fact  that  the  similarity- 
transformation  EI  =  (—  1,  —1)  is  present  in  the  group. 
Thus,  S  =  (abc)  may  transform  W\  into  Wz  or  into  WzEi, 
etc.  There  results  four  possible  forms  for  S,  all  of  which 
are  present  in  G  if  one  of  them  is.  We  shall  choose  the 
following  form : 

-l+i     -l+i 
2  2 

l+i     -l-i 
2  2 

(C)  Tetrahedral  group. 

G':  generated  by  (abc)  and  (ab)(cd); 

G:    a    group    (primitive;     §  60)    of    order    120 

generated  by  the  transformations  Wi  and 

S  above. 

The  rotations  of  the  octahedron  include  those  of  the 
tetrahedron  (abc)  and  (ab)(cd)  if  here  a,  b,  c,  d  represent 
each  a  pair  of  opposite  faces.  To  the  list  of  generating 
rotations  we  now  add  one,  U  say,  having  the  same  axis 
as  Wi,  but  being  of  period  4:  (acbd),  or  U2=Wi.  The 
corresponding  transformation  is  easily  determined  from 
this  last  equation.  We  find  that  it  has  the  canonical 
form 

U 


v/2'     1/2  A 


72  FINITE  COLLINEATION  GROUPS 

(D)  Octahedral  group. 

G'  :  generated  by  (abc)  and  (acbd)  ; 
G:    a  group  (primitive)  of  order  24<£,  generated  by 
S  and  U  above. 

58.  The  icosahedron.  Counting  the  rotations  of  an 
icosahedron  we  find 

1  axis  of  period  1  (the  identity), 
15  axes  of  period  2, 
20  axes  of  period  3,  and 
24  axes  of  period  5, 

making  60  in  all,  the  order  of  G'. 

A  Sylow  subgroup  of  order  4  (§  36)  must  be  repre 
sented  by  three  mutually  perpendicular  axes  of  period  2 
(§55);  moreover,  two  distinct  subgroups  of  order  4 
can  have  no.  axis  of  period  2  in  common,  since  otherwise 
such  an  axis  would  be  of  higher  period.  Hence,  the  15 
axes  of  period  2  must  belong  to  5  subgroups  of  order  4. 

It  is  readily  observed  that  no  rotation  of  G'  can  trans 
form  each  of  these  subgroups  into  itself.  It  follows  that 
G'  can  be  written  as  a  substitution  group  on  5  letters 
a,  6,  c,  d,  e,  simply  isomorphic  with  Gf  (§46).  The  20 
rotations  of  period  3  are  represented  by  all  the  cycles  on 
three  letters;  that  is,  the  substitution  group  in  question 
is  the  alternating  group  on  5  letters  (§  49). 

Now,  this  group  is  generated  by  the  following  operators 
(§50):  Fi,  Fz,  F3  (corresponding  to  the  substitutions 
(abc),  (ab)(cd),  (ab)(de)),  which  satisfy  the  relations 


For  the  corresponding  transformations  of  G,  we  may 
evidently  take  respectively  S,  W\  above,  and  a  new 
transformation  V  which  fulfils  the  conditions 


=  E  or  Ei        V2  =     or 


LINEAR  GROUPS  IN  TWO  VARIABLES  73 

The  first  and  last  ambiguities  fall  away,  since  of  neces 
sity  V*=(SV)*  =  Ei  (cf.  §51,  3°);  and  by  using  VEitt 
necessary  instead  of  V  we  may  take  (WiV)3  =  E.  We 
then  find 


v= 

~2 

where 

l-J/5        =  1  +  1/5 


(E)  Icosahedral  group. 

G':  generated   by    certain   rotations   of   periods 

3,  2,  2,  corresponding  to  the  substitutions 

(o6c),  (ab)(cd),  (ab)(de)', 
G:    a  group  (primitive)  of  order  60<£  generated 

by  S,  Wi,  and  V  above. 

The  group  (E)  may  be  given  the  following  useful  form.  Cor 
responding  to  the  substitutions  S'  =  (abcde),  t/'  =  (od)(6c),  T'  = 
(ab)(cd)  of  the  alternating  group  on  five  letters,  a  set  of  generators 


are  constructed  from  the  relations  S'5  =  E,  U'-1S'U'  =  S'-lt  T'2=Ei, 
U'-1T'U'  =  T'  or  =T'Ei.  The  transformation  Sf  is  at  the  outset 
written  in  canonical  form  ;  during  the  subsequent  determinations  of 
U'  and  T'  we  simplify  undetermined  coefficients  as  much  as  possible 
by  suitable  changes  of  variables.  The  two  types  of  groups  (E)  here 
given  are  equivalent  (§51). 

59.  Jordan's  process.*  We  shall  in  conclusion  give 
an  outline  of  Jordan's  method  for  determining  the  linear 
groups  in  two  variables. 

*  Jordan,  op.  cit.,  in  footnote  to  §  52;  Valontinor,  op.  cit.,  in  the  same 
footnote. 


74  FINITE  COLLINEATION  GROUPS 

1°.  It  can  be  proved  without  difficulty  that  if  two 
different  abelian  groups  in  two  variables  have  in  common 
a  transformation  S  which  is  neither  E  nor  E\,  then  the 
transformations  of  the  two  groups  are  mutually  commuta 
tive,  so  that  they  both  belong  to  a  single  abelian  group. 
(Writing  S  in  canonical  form,  we  find  that  a  transforma 
tion  which  is  commutative  with  it  has  the  canonical 
form  also.) 

2°.  Let  KI  be  an  abelian  group  written  in  canonical 
form.  Then  if  T  transforms  KI  into  itself  (T~lKiT  =  K1; 
cf.  §  30)  and  is  not  commutative  with  each  of  the  trans 
formations  of  Kit  we  can  readily  prove  that  it  must  inter 
change  the  variables  of  K\  (i.e.,  T  has  the  form  , 

3°.  Now  let  G  be  a  linear  group  in  two  variables  and 
of  order  g<j>.  By  1°,  we  can  sort  its  transformations  into 
distinct  abelian  subgroups  K\,  Kz,  .  .  ,  Km,  of  orders 
&i<£,  kz<f»t  .  .  ,  fcm<£,  such  that  no  transformation,  except 
E  and  EI,  occurs  in  two  distinct  subgroups.  Hence  we 

have  0*  =  <H-(fci- l)*  +  (k- !)<#>+  -  -  +(fcw-l)*;  (It 
is  observed  that  E  and  EI  are  counted  once  each  in  the 
term  <£  in  the  right-hand  member,  but  not  in  any  of  the 
numbers  (ki—  !)<£,  .  .  .) 

4°.  The  various  groups  K\,  K2,  .  .  ,  Km  may  be 
distributed  into  conjugate  sets.  Thus,  if  KI,  K2,  .  .  ,  K^ 
make  up  one  such  set,  the  orders  ki<f>,  &2<£,  .  .  are  equal, 
and  the  subgroup  H'  which  contains  K\  invariantively 
(cf.  §  30)  is,  by  2°,  of  order  2&i<£  or  ki<j>}  according  as  there 
is,  or  is  not,  a  transformation  of  type  T  in  G  interchanging 
the  variables  of  KI.  Hence,  if  the  subgroups  are  arranged 
in  conjugate  sets  as  suggested  above,  we  get 


LINEAR  GROUPS  IN  TWO  VARIABLES  75 

which  reduces  to 

m  i 

— 


2k" 

This  diophantine  equation  is  now  shown  to  have  a 
finite  number  of  solutions  for  the  integers  g,  k',  k",  .  .  , 
and  by  the  aid  of  these  solutions  the  various  groups  G 
may  be  determined  without  much  trouble. 

EXERCISES 

1.  Construct  the  linear  transformations  in  the  variables  X,  Y, 
Z  (cf.  §  54)  corresponding  to  the  generating  rotations  Wi,  W2,  W3, 
S,  U  and  F,  §§  57-58. 

Hence  show  that  the  groups  (C)  and  (D)  have  the  monomial 
form  (§  60)  when  written  in  the  variables  X,  Y,  Z. 

2.  Verify  the  diophantine  equation  (1)  for  the  groups  (A)  to 
(E).     (Hint:    to  each  abelian  subgroup  of  G  corresponds  a  single 
axis  of  rotation  of  (/'.) 


CHAPTER  IV 


ADVANCED  THEORY  OF  LINEAR  GROUPS* 
A.      ON   IMPRIMITIVE   GROUPS   AND   SYLOW   SUBGROUPS 

60.  Primitive  and  imprimitive  groups.  Let  us  sup 
pose  that  a  group  G  contains  not  only  transformations 
of  the  type  given  in  the  example  in  §  14,  but  also  some  of 
type 

p       q       t       v 

—  q  —p  —v    —t 
u      w      r      s 

—  w  —u  —s   —r 

which  upon  the  change  of  variables  employed  in  §  14 
becomes 

0        0      p-q  t-v 

0        0     u  —  w  r—s 


p+q  t+v 
u+w r+s 


then  we  say  that  G  is  imprimitive,  under  the  assumption 
that  it  is  transitive. 

In  general,  a  transitive  group  G,  in  which  the  variables 
(either  directly  or  after  a  suitable  choice  of  new  variables) 
can  be  separated  into  two  or  more  sets  Y\,  .  .  ,  Yk,  such 
that  the  variables  of  each  set  are  transformed  into  linear 
functions  of  the  variables  of  the  same  set  or  into  linear 
functions  of  the  variables  of  a  different  set,  is  said  to  be 
imprimitive.  If  such  a  division  is  not  possible,  the  group 

*  Wo  observe  the  rules  laid  down  in  §  51,  with  the  exception  that  the 
transformations  discussed  in  §§  60-62  are  not  restricted  to  be  of  determi 
nant  unity. 

76 


ADVANCED  THEORY  OF  LINEAR  GROUPS         77 

is  primitive.     The  sets   FI,   .    .   ,  F*  are  called  sets  of 
imprimitivity . 

THEOREM  1.  The  n  variables  of  an  imprimitive  linear 
group  G  may  be  so  chosen  that  they  break  up  into  a  certain 
number  of  sets  of  imprimitivity  YI,  F2,  .  .  ,  F&  of  m 
variables  each  (n  =  km),  permuted  among  themselves  accord 
ing  to  a  transitive  substitution  group  (§  44)  K  on  the  k 
letters  FI,  .  .  ,  Yk,  isomorphic  with  G.  To  the  subgroup 
of  K  which  leaves  the  letter  YI  unaltered  (§  45)  there  cor 
responds  a  subgroup  of  G  which  is  primitive  as  far  as  the 
variables  of  the  set  YI  are  concerned. 

If  m=l,  k  =  n,  then  G  is  said  to  have  the  monomial 
form  or  to  be  a  monomial  group. 

Proof. — The  transformations  Si,  &,  .  .  ,  Sh  in  G 
which  transform  the  variables  of  each  set  into  linear 
functions  of  the  variables  of  that  same  set  (there  is  at  least 
one  such  transformation,  namely  E),  form  a  group; 
moreover,  if  V  is  any  transformation  in  G  which  permutes 
the  sets  in  a  certain  way,  then  SiV,  .  .  ,  ShV,  and  no 
others,  permute  them  in  the  same  way.  If  we  therefore 
associate  with  each  such  system  of  h  transformations  a 
substitution  on  the  letters  FI,  .  .  ,  F*,  indicating  the 
manner  in  which  V  permutes  the  corresponding  sets  of 
imprimitivity,  we  obtain  a  substitution  group  K,  to  which 
G  is  (h,  1)  isomorphic  (§  32). 

Since  G  is  a  transitive  linear  group  (§  14),  K  must  be 
a  transitive  substitution  group  (§  44).  Hence,  K  con 
tains  k—  1  substitutions  T2)  T3,  .  .  ,  T^  which  replace 
FI  by  F2,  F3,  .  .  ,  Ffc  respectively  (§  45).  Let  us 
select  k—l  corresponding  transformations  of  G  and 
denote  them  by  ¥2,  F3,  .  .  ,  Vk.  The  conditions  that 
their  determinants  must  not  vanish  is  found  to  imply 
that  the  sets  contain  an  equal  number  of  variables,  say 
ra,  so  that  n  =  km.  (A  simple  example  will  suffice  to  show 
this  clearly,  say  n  =  4,  k  =  2.) 


78  FINITE  COLLINEATION  GROUPS 

There  is  in  K  a  subgroup  KI  whose  substitutions  leave 
YI  unaltered  (§45).  This  subgroup,  together  with  the 
k—l  substitutions  corresponding  to  V%,  .  .  ,  Vk  will 
generate  K.  Correspondingly,  G  is  generated  by  Vz, 
.  .  ,  Vk  and  that  subgroup  G\  whose  transformations 
replace  the  m  variables  of  the  set  YI  by  linear  functions 
of  the  same  variables,  however  they  may  permute  the 
remaining  sets.  Let  us  fix  our  attention  upon  just  that 
portion  of  each  of  the  transformations  of  G\  which 
affect  only  these  m  variables,  and  which  may  be  looked 
upon  as  forming  the  transformations  of  a  linear  group 
[(jj.  We  shall  proceed  to  prove  that  if  [G\]  is  not 
primitive,  then  the  variables  in  G  may  be  changed  in  such 
a  way  that  the  number  of  new  sets  of  imprimitivity  is 
greater  than  k. 

Assuming  then  that  [Gi]  is  not  primitive,  its  m  vari 
ables  are  found  to  break  up  into  at  least  two  subsets  of 
intransitivity  or  imprimitivity  (either  directly  or  after  a 
change  of  variables) .  For  the  sake  of  definiteness  we  shall 
assume  just  two  such  sets,  Y{,  Y".  New  variables  will 
now  be  introduced  within  the  sets  F2,  .  .  ,  Yk  such  that 
V2  will  replace  Y[,  T{  by  two  distinct  sets  Y'z,  Y'i  in  F2; 
and  so  on  for  each  of  the  transformations  V3,  V*,  .  .  , 
Vk.  Let  us  temporarily  write  (Yr)T—Y8  to  represent 
the  phrase  "  T  transforms  the  variables  of  the  set  Yr  into 
linear  functions  of  the  variables  of  the  set  Y8";  and 
(Y'r,  Y'J)T=(Y'S,  F'/)  to  represent  the  phrase  "T  trans 
forms  the  variables  of  any  subset  of  Yr  into  linear  func 
tions  of  the  variables  of  a  subset 'of  F8"  (that  is,  either 
(Y'r)T=Y'.,  (rr')!T=F'/;  or  (F;)T=F'/,  (Y^T=Y'8). 
Then  the  properties  of  the  2k  subsets  relative  to  the 
transformations  F2,  .  .  ,  Vk  may  be  stated  in  the  form : 
(FJ,  FY)F«=(F{,  r/),  or  its  equivalent  form:  (FJ,  F'/)  Fi1 
=  (Y{,  F'/);  «  =  2,  .  .  ,  k. 

It  remains  for  us  to  prove  that  any  transformation  W 
in  G  will  permute  these  subsets  among  themselves;  that  is, 


ADVANCED  THEORY  OF  LINEAR  GROUPS         79 

assuming  (Fp)Tf=Fg,  then  we  have  also  (Yp,  Y'P)W  = 
(Yq,  Yq).     To  show  this,  consider  the  transformation 


This  transformation  belongs  to  GI,  since  it  transforms  the 
set  YI  into  itself: 


Hence,  by  assumption,  (Y[,  Y'{)T=(Y[,  F'/);  and,  since 
VpW=TVq,  so  that  (71,  Y'{)VpW=(Yi,  Y'{)TVq,  we 
derive  (Y'p,  Yp)W=(Yi,  Y'q),  what  we  set  out  to  prove. 
Accordingly,  if  [Gi]  is  not  primitive,  we  can  change  the 
n  variables  of  G  so  as  to  increase  the  number  of  sets. 
This  process  may  be  continued  until  the  sets  contain  just 
one  variable  each,  or  until  we  get  a  group  [Gi]  which  is 
primitive.  Hence  the  theorem. 

EXERCISES 

1.  Prove  that  there  is  only  one  type  of  imprimitive  groups  in 
three  variables,  namely  the  monomial  type.     Prove  also  that  there  is 
a  single  non-monomial  type  of  imprimitive  groups  in  four  variables. 

2.  Prove  that  the  group  generated  by  S  =  (  —  1,  1,  —1)  and  T: 
xl=-i(x[+2xi+xi)/2,    Xt  =  (-x[+xi)/2,     xt  =  i(x[-2xL+xb/2,    is 
imprimitive.     (Hint:  By  Exercise  1,  the  group  must  be  monomial. 
Hence,  if  the  new  variables  are  x,  y,  z,  the  function  xyz  is  a  relative 
invariant  (  §  88)  of  both  S  and  T.) 

61.  Sylow  subgroups. 

LEMMA.  A  linear  group  G  which  contains  an  invariant 
abelian  subgroup  H  not  composed  entirely  of  similarity- 
transformations,  is  either  intransitive  or  imprimitive. 

Proof.  —  Write  H  in  canonical  form.  The  variables 
can  then  be  arranged  in  sets  Xi,  Xz,  .  .  .  ,  having  the 
property  that  a  transformation  of  H  affects  all  the  vari 
ables  of  any  one  set  by  the  same  constant  factor. 

To  illustrate,  let  H  be  generated  by  the  transformations 


i  =   ai,  O!,  alt   O!,  aj 
r,  =  (j8,,  ft,  ft,  ft,  ft)  (ft*  A). 

Here  we  have  three  sets:   Xi  =  (xlt  x2),  X2  =  (x3,  z4),  X3  =  (x6). 


80  FINITE  COLLINEATION  GROUPS 

We  shall  for  the  moment  write  the  phrase  "  T  trans 
forms  the  variables  of  the  set  Xp  into  one  and  the  same 
constant  multiple  of  themselves"  symbolically:  (XP)T  = 
c(Xp).  Thus,  in  the  illustration  given,  (X2)T2  =  c(X2), 
the  constant  multiple  being  /32.  More  generally,  if  X' 
denotes  an  aggregate  of  certain  linear  functions  yiy  y2,  .  . 
of  the  variables  of  the  group,  the  equation  (X')T  =  c(X') 
implies  that  (yi)T  =  ayi,  (y2)T  =  ay2,  .  .  ,  a  being  a 
constant,  the  same  for  every  function  y\,  y2)  .  .  .  Then 
if  (X')T  =  c(Xr)  for  every  transformation  T  of  H,  it  follows 
that  X'  cannot  contain  variables  from  two  or  more  distinct 
sets  X\j  X2)  .  .  .  For,  otherwise  at  least  one  transforma 
tion  in  H  would  affect  some  of  the  letters  involved  by  one 
constant  factor,  and  others  by  a  different  constant  factor. 

Now  let  S  and  T  be  any  transformations  from  G  and 
H  respectively.  The  variables  of  a  given  set  Xp  are  by 
S  transformed  into  linear  functions  of  the  variables  of  G, 
forming  an  aggregate  which  we  shall  denote  by  X'. 
Since  H  is  invariant  under  G,  the  transformation  V  = 
STS~l  belongs  to  H,  and  we  have  (Xp)  V  =  c(Xp) .  Hence, 

(XP)ST=(XP)VS,  or  (X')T  =  c(Xf). 

Accordingly,  the  linear  functions  of  Xr  must  contain 
variables  from  only  one  set  of  H;  in  other  words,  S  trans 
forms  the  variables  involved  in  Xp  into  linear  functions 
of  the  variables  of  some  one  set  of  H.  The  sets  X\, 
X2)  .  .  are  therefore  permuted  among  themselves  by  S, 
and  the  lemma  is  established. 

THEOREM  2.  A  linear  group  whose  order  is  the  power 
of  a  prime  number  can  be  written  as  a  monomial  group  by  a 
suitable  choice  of  variables  Xi,  x2,  .  .  ,  xn;  that  is,  its 
transformations  have  the  form:* 

xs  —  astx't , 

*  First  given  by  the  author  in  Transactions  of  the  American  Mathe 
matical  Society,  V  (1904),  313-14;  VI  (1905),  232;  see  also  Burnside, 
Theory  of  Groups  of  Finite  Order.  2d  ed.,  Cambridge,  1911,  p.  352. 


ADVANCED  THEORY  OF  LINEAR  GROUPS         81 

where  s  and  t  run  through  the  numbers  1,  2,  .  .  ,  n,  though 
not  necessarily  in  the  same  order. 

Proof. — 1°.  A  group  P  whose  order  is  the  power  of  a 
prime  number  p  is  either  abelian  or  it  contains  an  invariant 
abelian  subgroup  Q  whose  transformations  are  not  sepa 
rately  invariant  under  P  (§  35).  In  the  first  case  the  the 
orem  follows  from  Theorem  10,  §  22.  In  the  second  case, 
Q  can  be  written  in  canonical  form,  and  P  is  intransitive 
or  imprimitive  by  the  above  lemma.  If  the  theorem  is 
true  for  a  transitive  group,  it  is  evidently  true  for  an 
intransitive  group;  hence  we  need  merely  consider  the 
case  where  P  is  imprimitive. 

2°.  By  Theorem  1,  §  60,  P  is  monomial  unless  there 
is  a  group  [Pi]  which  is  primitive  in  the  m  variables  of  a 
set  YI.  But  the  latter  alternative  is  impossible  by  1°, 
since  the  order  of  [PJ  is  again  a  power  of  p,  being  a  factor 
of  such  a  number.  Hence  the  theorem. 

COROLLARY.  A  linear  group  P  in  n  variables,  whose 
order  is  a  power  of  a  prime  number  p  which  is  greater  than 
n,  is  abelian. 

Proof. — Write  P  in  monomial  form.  Then  any  trans 
formation  T  which  does  not  have  the  canonical  form  will 
permute  the  variables  Xi,  x%,  .  .  ,  xn  in  a  certain  manner, 
indicated  by  a  substitution  S  on  these  letters.  If  the 
order  of  S  is  k,  that  of  T  is  k  or  a  multiple  of  k.  Now, 
the  order  of  T  is  a  power  of  p  (§  28,  Corollary) ;  it  follows 
that  k  is  a  power  of  p.  But  this  is  impossible  since  k 
is  n  or  a  product  of  numbers  all  less  than  n  (cf .  Exercise  4, 
§41),  and  none  of  the  prime  factors  involved  can  be  p. 
Hence,  every  transformation  of  P  has  the  canonical  form, 
and  this  group  is  accordingly  abelian. 

EXERCISE 

If  pb  is  the  highest  power  of  p  which  divides  n\,  prove  that  a 
Sylow  subgroup  of  order  pa  in  n  variables  contains  an  invariant 
abelian  subgroup  of  order  pa-b  at  least,  if  a>b. 


82  FINITE  COLLINEATION  GROUPS 

62.  Theorem  3.  A  linear  group  in  n  variables  and 
of  order  g  =  g'paqbrc  .  .  ,  where  p,  q,  r,  .  .  are  different 
primes  all  greater  than  n+1,  contains  an  abelian  subgroup 
of  order  paqbrc.  .  . 

To  prove  this  theorem  by  complete  induction,  we 
assume  it  true  for  any  group  in  less  than  n  variables  and 
for  any  group  in  n  variables  whose  order  is  less  than  g. 
That  the  theorem  is  then  true  for  an  intransitive  group 
in  n  variables  will  be  shown  below  (1°).  There  is  left  the 
case  of  a  transitive  group  G  of  order  g  in  n  variables.  It 
may  be  assumed  that  the  determinants  of  the  transforma 
tions  of  G  are  all  unity  (cf.  2°  below). 

At  the  outset  we  anticipate  a  theorem  given  later 
(cf.  Exercise  2,  §  90).  From  this  it  follows  that  G  con 
tains  an  abelian  subgroup  HI  of  order  pq.  This  subgroup 
contains  a  transformation  S  of  order  p  which  is  also 
contained  in  a  Sylow  subgroup  P  of  G  of  order  pa  (§  39). 
The  groups  HI  and  P  being  abelian,  S  is  invariant  under 
both,  and  will  therefore  also  be  invariant  under  the  group 
K  generated  by  Hi  and  P,  whose  order  is  divisible  by  paq. 
Now,  S  cannot  be  a  similarity-transformation  (by  2°; 
otherwise  its  determinant  could  not  be  unity) ;  it  follows 
that  K  is  intransitive  (cf.  proof  of  Theorem  10,  §  22), 
and  must  therefore,  by  1°,  contain  an  abelian  subgroup 
H2  of  order  paq. 

Again,  this  subgroup  Hz  and  a  certain  Sylow  subgroup 
Q  of  G  of  order  qb  have  in  common  a  transformation  T 
of  order  q,  which  must  be  invariant  under  both.  We  find, 
by  the  process  above,  an  abdian  subgroup  H'  of  order 
paqb.  In  the  same  way  we  obtain  abelian  subgroups 
H"}  .  .  of  orders  p°rc,  .  .  . 

Now,  the  subgroups  of  order  pa  form  a  single  conjugate 
set  within  G.  We  may  therefore  select  such  conjugates 
of  H",  H'",  .  .  ,  that  these  conjugates  have  in  common 
with  H'  the  group  P.  The  group  generated  by  them  will 
be  intransitive,  the  operators  of  P  being  invariant,  and 


ADVANCED  THEORY  OF  LINEAR  GROUPS         83 


its  order  will  be  a  multiple  of  paqbrc  .  .  .     The  theorem 
follows  by  1°. 

1°.  Consider  an  intransitive  group  G  in  two  sets  of  intransi- 
tivity,  the  corresponding  component  groups  being  designated  G' 
and  G".  In  G'  we  find,  by  assumption,  an  abclian  subgroup  K' 
of  order  k'  =  paqPry  .  .  ,  which  we  shall  write  in  canonical  form. 
Let  the  identity  in  G'  correspond  to  a  group  G'{  in  G",  of  order  g"; 
the  order  of  G  will  be  the  product  of  the  orders  of  G'  and  G"  (§32). 
Now,  in  G'i  we  find  an  abelian  subgroup  K"  of  order  p^qT*  .  .  ; 
writing  this  in  canonical  form  we  see  that  the  group  generated  by 
K'  and  K"  is  the  group  sought.  * 

2°.  If  the  determinants  of  the  transformations  of  G  are  not 
all  unity,  we  construct  a  group  G\  by  the  method  of  §  12  whose 
transformations  have  this  property.  If  then  we  find  in  Gi  an 
abelian  group  of  order  paqPry  .  .  ,  we  evidently  have  a  correspond 
ing  group  in  G,  also  abelian.  The  latter  may  contain  similarity- 
transformations  whose  orders  are  prime  to  paqbrc  .  .  •  but  these 
transformations  can  easily  be  disposed  of  (§  34). 

EXERCISES 

1.  If  n-f  lisa  prime,  andif  theorderof  Gisg  =  g'(n-{-l)tpaqb  .    .  t 
where   t>l,   and   p,   q,    .    .    are   primes  greater  than  n  +  1,  prove 
that  there  is  in  G  an  abelian  subgroup  of  order  (n-\-\yp<*q*>  .    .    . 

2.  It  follows  from  Theorem  3  that  a  group  in  n  variables  whose 
order  is  not  divisible  by  a  prime  factor  smaller  than  n+2  is  abelian. 
Prove  that  if  the  order  is  not  divisible  by  a  prime  factor  smaller 
than  n+1,  the  group  is  abelian. 

B.      ON   THE   ORDER   OF   PRIMITIVE   GROUPS 

63.  The  remainder  of  this  chapter  will  be  devoted  to 
the  discovery  of  limits  to  the  magnitude  of  the  prime 
factors  that  may  enter  the  orders  of  primitive  groups 
(§§  63-64),  to  the  highest  admissible  powers  of  the  prime 
factors  under  certain  conditions  (§§  65-68),  and  to  the 
orders  of  abelian  subgroups  in  general  (§§  69-73).  The 
chapter  closes  with  a  discussion  of  the  order  of  a  primitive 
group  (§74). 

THEOREM  4.  No  prime  number  p>7  can  divide  the 
order  of  a  primitive  group  G  in  three  variables. 


84  FINITE  COLLINEATION  GROUPS 

The  method  of  proof  consists  in  showing  that  if  the 
order  g  contains  a  prime  factor  p>7,  then  G  is  not  primi 
tive.  We  subdivide  this  process  into  four  parts  as  fol 
lows:  1°  proving  the  existence  of  an  equation  F  =  Q, 
where  F  is  a  certain  sum  of  roots  of  unity;  2°  giving  a 
method  for  transforming  such  an  equation  into  a  con 
gruence  (mod  p) ;  3°  applying  this  method  to  the  equation 
F  =  0;  4°  deriving  an  abelian  self-conjugate  subgroup  P 
of  order  pk. 

1°.  The  order  g  being  divisible  by  p,  G  contains  a 
Sylow  subgroup  of  order  pa  and  therefore  a  transformation 
S  of  order  p.  We  choose  such  variables  that  S  has  the 
canonical  form 

Two  cases  arise :  two  of  the  multipliers  are  equal,  say  ot  = 
03,  or  they  are  all  distinct.  They  cannot  all  be  equal, 
since  a\  =  1  and  a?  =  1  imply  a:  =  1 ;  whereas  S  is  not  the 
identity.  Of  the  two  cases  we  shall  treat  the  latter  only; 
the  method  would  be  the  same  in  the  former  case,*  and 
the  result  as  stated  in  Theorem  4  would  be  the  same. 
Selecting  from  G  any  transformation  V  of  order  p: 

[ai  bi  Cil 
02  62  c2    , 
as  &s  c3J 

we  form  the  products  VS,  VS2,  VS*.  Their  character 
istics  (§  23)  and  that  of  V  will  be  denoted  by  [VS],  .  .  .  , 
[V],  and  we  hive 

[V]     =ai 
[VS]  =a* 


*  The  congruence  (7)  would  hero  be  of  the  first  degree  in  /*. 


ADVANCED  THEORY  OF  LINEAR  GROUPS         85 

We  now  eliminate  ai,  62,  c3,  from  these  equations,  obtaining 
[V]        111 

[VS\        Oil     0.2    "3 


(2) 


a?    a2 


=  0. 


Expansion  and  division  by  (a!— a2)(a2— a3)(a3  —  ax)  gives  us 
(3)  [V&]+K[V]+L[VS\+M[V&]  =  Q, 

K,  L,  M  being  certain  polynomials  in  ab  a2,  a3  (§  132), 
with  the  general  term  of  type  afa^af.  Since  al}  a2,  a3  are 
powers  of  a  primitive  pth  root  of  unity  a  (§  133),  the 
quantities  K,  .  .  .  are  certain  sums  of  powers  of  a. 
Moreover,  the  characteristics  [V],  .  .  .  are  each  the 
sum  of  three  roots  of  unity  (Exercise  8,  §  6).  If,  there 
fore,  the  products  in  (3)  were  multiplied  out,  there  would 
result  an  equation  of  the  kind  discussed  in  §  133,  6°. 
The  various  terms  could  therefore  be  rearranged  in  sets 
as  explained  in  that  paragraph,  which  gives  us  an  equation 
of  the  form 


A,  B,  C,  .  .  .  being  certain  sums  of  roots  of  unity; 
a,  /?,  y,  .  .  .  primitive  roots  of  the  equations  xp=l, 
xq=l,  xr=l,  .  .  .  respectively;  and  p,  q,  r,  ... 
different  prime  numbers. 

The  coefficients  A,  B,  C,  .  .  .  may  be  put  into 
certain  standard  forms.  Thus,  any  root  c4=l  occurring 
in  any  of  these  sums  will  be  assumed  to  be  resolved  into 
factors  of  prime-power  indices  (§  133,  4°):  c  =  cpcy€r  .  .  , 
the  root  «p  being  of  index  pm,  cg  of  index  qn,  etc.  Further 
more,  within  A  any  root  cp  will  be  assumed  to  be  either 
unity  or  a  root  whose  index  is  divisible  by  p2.  For,  if 


86  FINITE  COLLINEATION  GROUPS 

tp  were  of  index  p,  say  €p  =  afc,  we  could  put  1  in  its  place, 
since 

a2_|_     ^     f     ^     _f_ap-l)=a*_J_a*+l+a*+2_|_     ^     ^     ^ 


by  means  of  the  relation  a?  =  1.  Likewise  we  assume  that 
any  root  eg  within  J5  is  either  equal  to  unity  or  is  a  root 
whose  index  is  divisible  by  q2;  and  so  on. 

To  illustrate,  take  p  =  2,  <?  =  3,  and  let  i  be  a  root  of  index  4 
(12=  _  i)  and  r  a  root  of  index  9  (r3  =  w,  w3  =  l).  Then  the  standard 
form  for  the  expression 

(5)  (-l'a;-l)(l 

would  be 


2°.  We  shall  now  make  certain  changes  in  the  values 
of  the  roots  in  the  equation  (4).  First  we  put  0  for  every 
root  €g,  CT,  .  .  .  whose  index  is  divisible  by  the  square 
of  a  prime  other  than  p2  (as  r5  in  the  example  above), 
leaving  undisturbed  the  roots  whose  indices  are  not  divis 
ible  by  such  a  square,  as  a,  a2,  .  .  .  ,  /?,  .  .  .  .  Al 
though  these  changes  will  turn  the  quantities  A,  B,  .  .  . 
into  certain  new  sums  A',  Bf,  .  .  .  .  ,  the  equation 
(4)  is  still  true,  since  the  vanishing  factors  l-f-a-f-a2-|- 
.  .  .  +aP~1,  etc.,  have  not  been  affected. 

Next  we  put  0  in  place  of  q  —  2  of  the  roots  ft  /32, 
.  .  .  ,  fiq~l,  and  —1  for  the  remaining  root,  thus  chan 
ging  B'(l+P+!3*+  .  .  .  +/?*-1)  into  £'(1+0+0+ 
.  .  .  +(—!)),  so  that  this  product  still  remains  equal 
to  zero.  Similarly  we  put  0  in  place  of  r—  2  of  the  roots 
y,  y2,  .  .  .  ,  yr~l,  and  —1  for  the  remaining  root,  and  so 
on.  Proceeding  thus,  we  shall  ultimately  change  (4) 
into  an  equation  of  the  form 


where  A"  contains  roots  of  the  form  =*=cp  only. 


ADVANCED  THEORY  OF  LINEAR  GROUPS          87 

Finally,  we  put  1  in  the  place  of  every  root  a,  a2, 
.  .  .  ,  aP"1,  as  well  as  every  root  ep.  The  left-hand 
member  may  then  no  longer  vanish,  but  will  in  any 
event  become  a  multiple  of  p. 

The  final  value  of  the  expression  (5)  would  be  (w  +  l)(l-|-l)  = 
2  or  0,  according  as  w  is  replaced  by  0  or  —1. 

Notation  1. — Any  expression  N  which  is  a  sum  of 
roots  of  unity  changed  in  the  manner  described  above, 
shall  be  denoted  by  N'p. 

3°.  We  shall  now  study  the  effect  of  these  changes 
upon  the  left-hand  member  of  (3).  Each  of  the  char 
acteristics  [VS],  .  .  .  ,  [F$M],  being  the  sum  of  three 
(unknown)  roots  of  unity,  will  finally  become  one  of  the 
seven  numbers  0,  ±1,  =±=2,  ±3,  whereas  [V],  being  the 
sum  of  three  roots  of  index  1  or  p  (cf.  1°),  will  become  3. 
The  left-hand  member  of  (3)  will  thus  take  the  form 

(6)  [V&\i+3Kl+L&VSll+MaV&U, 

and  this  number  is  a  multiple  of  p,  by  2°. 

The  values  Kp,  Lp,  M'p  may  be  obtained  by  treating 
them  as  indeterminates  0/0.  Thus, 


K  = 


—  02)  (02—  i 

We  find  (cf.  §  132) 


0-2      a3 


0* 


and  if  we  substitute  in  (6)  and  multiply  by  p  —  I  we  obtain 
the  congruence 


(7)  [VS"]l=8p*+tfJi+v     (modp), 

s,  t,  v  being  certain  integers,  the  same  for  all  values  of  /x. 

We  finally  substitute  in  succession  ^  =  0,  1,  2,   .    .   , 

p—  1  in  the  right-hand  member  of  (7).     The  remainders 


88  FINITE  COLLINEATION  GROUPS 

(mod  p;  cf.  §  129)  should  all  lie  between  —3  and  +3 
inclusive,  the  interval  of  the  values  of  [VS^p.  Now,  each 
of  these  seven  remainders  can  correspond  to  only  two 
different  values  of  ^  less  than  p,  if  s  and  t  are  not  both 
=  0  (mod  p)  (§  130).  Hence,  there  will  correspond  to  each 
of  the  seven  remainders  at  most  14  different  values  of  /*, 
so  that  p  is  not  greater  than  14  unless  s=£=0.  Trying 
p=  13  and  p  =  11,  choosing  for  s,  t,  v  the  different  possible 
sets  of  numbers  <p  (the  problem  can  be  simplified  by 
special  devices),*  we  find  that  in  no  case  can  the  re 
mainders  all  be  contained  in  the  set  0,  ±1,  ±2,  ±3, 
unless  s=2=0.  Choosing  therefore  this  alternative  we 
get,  if  p>7, 

[V&]i=v         (mod  p). 

In  particular, 

v=[V\i  =  3         (modp), 


from  which  it  follows  that  [VS]P  =  3.  Again,  from  this 
equation  we  deduce  that  the  roots  of  [VS]  are  of  index  1 
or  p.  For  if  the  index  of  one  of  these  roots  were  divisible 
by  the  square  of  a  prime,  or  by  a  prime  different  from  p, 
then  the  changes  indicated  in  2°  could  be  made  at  the 
outset  in  such  a  way  that  0  or  —  1  would  take  the  place 
of  this  root.  But  then  [VS]P  would  be  one  of  the  numbers 
0,  ±1,  ±2,  -3. 

4°.  Accordingly,  the  product  VS  of  any  two  trans 
formations  both  of  order  p  is  a  transformation  of  order 
p  or  1.  The  totality  of  such  transformations  in  G, 
together  with  E,  will  therefore  form  a  group  P  (§  27). 
The  order  of  this  group  must  be  a  power  of  p  since  it 

*  Since  an  +b  runs  through  the  p  values  0,  1,  2  .....  p—  1  (mod  p) 
when  /*  does,  we  may  substitute  this  expression  for  M  in  the  right-hand 
member  of  (7)  and  select  the  constants  a,  b  so  that  this  member  takes  a 
simpler  form.  For  instance,  if  p  =11,  the  right-hand  member  of  (7)  may 
bo  reduced  by  this  substitution  to  one  of  the  forms  ±M2  +c;  M;  ore;  accord 
ing  as  8*0;  s=0,  <^0;  s=t=0  (mod  p).  When  p  =13  we  get  the  forms 
;  c. 


ADVANCED  THEORY  OF  LINEAR  GROUPS         89 

contains  no  transformation  whose  order  differs  from  p 
or  1.  Moreover,  P  is  invariant  under  G,  since  an  operator 
of  order  p  is  transformed  into  one  of  order  p  (Exercise 
2,  §  30).  Hence,  G  has  an  invariant  subgroup  P  of  order 
pk.  But  this  subgroup  is  abelian  (§61,  Corollary),  and 
therefore  G  is  intransitive  or  imprimitive  (Lemma,  §  61). 

64.  Case  of  n  variables.  Applying  the  process  indi 
cated  in  the  previous  paragraph  to  a  group  Ginn  variables, 
we  select  any  two  transformations  S,  V,  both  of  order  p, 
and  write  S  in  canonical  form 

S=(ai,   «2,    .     .     .    ,   «„). 

Assuming  that  the  multipliers  a1}  .  .  .  ,  an  are  all  dis 
tinct,  we  find  the  following  congruence  corresponding 
to  (7) 

(8)  [VS»]^snn-l+tnn-2+   .    .    .   (modp). 

There  are  at  most  2n-fl  different  values  that  [VS*]P  can 
take,  namely  0,  =*=!,  =fc2,  .  .  .  ,  =*=n;  and  each  cor 
responds  to  at  most  ra  — 1  of  the  p  values  of  /A:  0,  1,  2, 
.  .  .  ,  p—  1,  if  the  right-hand  member  is  not  merely 
a  constant.  It  follows  that  at  most  (2w+l)(n—  1)  of 
these  p  values  of  /*  can  satisfy  the  congruence  (8).  Ac 
cordingly,  if  p>(2n-fl)(n  —  1),  the  right-hand  member  of 
(8)  must  be  merely  a  constant;  and  then 

[V&]l=[VS\l=[V]l  =  n        (modp), 

from  which  we  find  [VS]p  =  n.  Now  we  deduce,  as  in 
4°,  §  63,  that  the  transformation  VS  is  of  order  p  or  1, 
and  from  this  that  G  has  an  invariant  abelian  subgroup 
of  order  pk  and  is  therefore  not  primitive.  Hence  the 

THEOREM  5.  The  order  of  a  primitive  linear  group 
in  n  variables  is  not  divisible  by  a  prime  number  which  is 
greater  than  (2n-\-l)(n—  1). 


90  FINITE  COLLINEATION  GROUPS 

In  the  case  n  =  4  this  limit  is  27.  But  we  may  try 
out  the  congruence  (8)  in  detail  for  the  primes  p  =  23, 
19,  .  .  ,  with  the  result  that  the  right-hand  member 
must  reduce  to  a  constant  when  p>!3.  For  p=l3  or 
11  the  process  fails,  as  congruences  of  the  form  (8)  exist 
for  these  primes  (for  instance,  when  p=l3,  (8)  is  satis 
fied  if  the  right-hand  member  is  the  function  2/x3). 

EXERCISES 

1.  No  transformation  of  variety  m  (cf.  §  65)  and  of  order  p, 
a  prime  greater  than  (2n-f-l)(ra  — 1),  can  belong  to  a  primitive 
group. 

2.  No  primitive  group  G  can  contain  a  transformation  of  order 
p2,  if  the  prime  p  is  greater  than  n  (§  67,  Corollary).  Hence  prove 
that  G  cannot  contain  a  subgroup  of  order  p2,  if  p=(2n  +  l)(n— 2) 
and  >2. 

65.  The  variety  of  a  linear  transformation  and  of  an 
abelian  group.    If  the  number  of  distinct  multipliers  of 
a  transformation  written  in  canonical  form  is  k,  we  say 
that  the  transformation  is  of  variety  k.     Thus,  the  trans 
formation  S=(l,  — 1,  — 1)  is  of  variety  2.     By  an  obvious 
extension  to  abelian  groups  written  in  canonical  form  we 
say  that  such  a  group  is  of  variety  k  if  its  variables  may  be 
separated  into  k  sets  in  the  manner  explained  in  §  61. 
The  group  H  of  the  illustration  in  that  paragraph  is 
accordingly  of  variety  3. 

Notation  2. — An  expression  N  which  is  the  sum  of  a 
certain  number  of  roots  of  unity,  in  which  every  root 
cp  is  replaced  by  1,  but  in  which  none  of  the  other  changes 
indicated  in  2°,  §  63,  are  carried  out,  will  be  denoted  by 
Np.  IfN  =  Q,  then  Np=0  (mod  p). 

66.  Theorem  6.     Let  a  group  G  contain  a  transforma 
tion  S  of  variety  k  and  order  pa<f>,  where  a>l  and  pa^kp; 
p  being  a  prime  number.     Then  there  is  an  invariant  sub 
group  Hp  in  G  (not  excluding  the  possibility  G  =  HP)  which 


ADVANCED  THEORY  OF  LINEAR  GROUPS          91 

contains  Spa~l.     Any  transformation  in  Hp,  say  T,  has 
the  property  expressed  by  the  following  congruence: 

(9)  [V]P~[VT]P         (modp), 

where  V  is  any  transformation  of  G. 

The  proof  follows  the  plan  of  the  proof  of  Theorem  4, 
§  63.  We  write  S  in  canonical  form,  and  construct  the 
products  VS,  VS2,  .  .  .  ,  VSk~l,  VR;  R  denoting 
Spa~l.  As  in  1°,  we  obtain  an  equation  corresponding 
to  (3): 

[VR]+K[V]+L[VS]+   .    .    . 


However,  the  changes  indicated  in  2°  are  not  carried  out 
except  that  1  is  put  for  every  root  ep  whose  index  is  a  power 
of  p  (cf.  Notation  2). 

All  the  coefficients  L,  .  .  .  ,  X  become  multiples 
of  p  by  this  change  (§132),  and  we  find  Kp=  —  1 
(mod  p).  Hence  finally, 

(10)  [VR]P-[V]P=Q     (modp). 

Now  consider  all  the  conjugates  Ri,  .  .  ,  Rh  to  R 
within  G.  They  generate  an  invariant  subgroup  Hp 
(Exercise  3,  §  31),  and  they  all  have  the  property  of  R 
as  expressed  by  (10),  since  they  fulfil  the  conditions  of  the 
theorem  (cf.  §  86,  3°).  Moreover,  any  transformation 
T  in  Hp  satisfies  the  congruence  (9),  since  such  a  trans 
formation  can  be  written  as  the  product  of  powers  of 
#1,  .  .  .  ,  Rh.  For  instance,  let  T  =  R1R2.  By  (10), 
we  have  [F/yp=[F]p,  and  [(VRi)Ri]p=[VRi]p  (mod  p). 
Hence, 

[F/yp=[F]p     (mod  p). 


67.  On  the  form  of  the  group  Hp.    Let  T  be  any 

transformation  of  Hp.     Then  by  (9), 

[£]„  =  «     (modp), 


92  FINITE  COLLINEATION  GROUPS 

and  therefore,  since  Tm  belongs  to  Hp  also, 
[Tm]p=n     (mod  p)  . 

Now  let  the  order  (say  t)  of  T  be  prime  to  p,  and  let 
us  assume  that  a  of  the  multipliers  of  T  are  equal  to  each 
other,  then  b  of  the  remaining  multipliers  equal,  and  so 
on.  That  is, 

m  =  a"i+&«2+   .    .    =  [T]P,        a+b+   .    .    =n, 
and 

[Tm]  =  0am:  +  bar?  +     .     .     =[Tm]p. 

Then  we  have  (§  133,  2°,  5°) 

t  t 


m  =  1  m  =  1 


=  0  or  =  nt,  according  as  a^  1  or  04.  =  1.  Hence,  we  must 
have  correspondingly  a=0  or  =n  (mod  p).  Like  results 
hold  for  the  other  numbers  6,  ... 

If  p>n,  the  congruence  a=0  (mod  p)  is  impossible 
unless  a  =  0.  It  follows  that  every  root  as  is  unity.  In 
other  words,  Hp  can  in  this  case  contain  no  transformation 
whose  order  is  prime  to  p.  The  order  of  Hp  is  therefore 
a  power  of  p,  and  G  is  not  primitive  (cf.  §  61).  Hence  the 

COROLLARY.  No  primitive  group  in  n  variables  can 
contain  a  transformation  of  order  p2,  if  p  is  a  prime  number 
greater  than  n. 

Remark.  —  The  group  Hp  as  defined  above  may  very  well  coin 
cide  with  G.  However,  it  may  be  shown  that  if  a  primitive  group  G 
contains  a  group  HP,  then  the  latter  (or  a  modified  form  of  it)  does 
not  make  up  the  whole  of  G,  and  n  must  be  divisible  by  p*. 

EXERCISE 

No  primitive  group  in  n  variables  can  contain  a  transformation 
'  of  order  p3,  if  p  is  a  prime  greater  than  n/2. 

*  Transactions  of  the  American  Mathematical  Society,  XII  (1911),  39-41. 


ADVANCED  THEORY  OF  LINEAR  GROUPS          93 

68.  Theorem  7.  Let  G  contain  two  commutative 
transformations  S=(ai,  .  .  ,  an)  and  T=(fti,  .  .  ,  /?„), 
satisfying  the  following  conditions: 

(A)  Their  orders  are  respectively  m$  and  p<f>,  p  being 
a  prime  number  which  is  not  a  factor  of  m; 

(B)  the  variety  k  of  S  is  equal  to  or  less  than  m; 

(C)  whenever  as  =  a.t,  then  shall  (3s  =  Pt  (the  converse  is 
not  implied). 

Under  these  conditions  G  has  an  invariant  subgroup  Hp 
which  contains  the  transformation  T. 

Proof. — Let  F  =  [as<]  be  any  transformation  of  G, 
and  let  the  products  VS,  VS2,  .  .  ,  VSk~l,  VT  be  con 
structed.  Eliminating  an,  022,  .  .  ,  ann  from  the  equa 
tions  F  =  an+022+  .  .  (cf.  (1),  §  63),  we  obtain  an 
equation  corresponding  to  (2),  with  the  elements  [F], 
[VS],  .  .  ,  [F/S*-1],  [VT]  in  the  first  column.  We  now 
write  1  for  every  root  cp  whose  index  is  a  power  of  p, 
before  expanding  the  determinant.  The  following  changes 
will  take  place:  the  elements  in  the  first  column  [F]  .  . 
become  [V]p  .  .  (cf.  Notation  2,  §  65),  and  the  elements 
of  the  last  row*  excepting  [FT]P,  all  become  unity.  Sub 
tracting  finally  the  last  row  from  the  first  and  expanding, 
we  get 

((V]P-[VT]P)D=0    (modp), 

D  being  the  product  of  the  differences  of  the  k  distinct 
roots  among  alt   .    .   ,  an. 

Now,  since  Sm  is  the  identity  or  a  similarity- 
transformation,  a^=  .  .  =a™.  Hence,  there  is  a 
rationalizing  factor  of  as—at  such  that  the  product  is 
m  or  a  factor  of  m.  (For,  put  at/as  =  0)  say,  a  root  of 
index  mi  (a  factor  of  m),  and  we  have  (§  133,  5°) 
(1-0)  .  .•(l-0m«-1)=mi*).  The  product  in  question 

*Thc  value  of  lira  - — ^-  . 


94  FINITE  COLLINEATION  GROUPS 

is  therefore  a  number  prime  to  p.  It  follows  that  the 
product  of  D  and  a  certain  rationalizing  factor  is  an  integer 
N,  prime  to  p.  The  resulting  congruence  may  now  be 
multiplied  by  an  integer  N'  such  that  NN'=1  (mod  p, 
§  131),  and  we  obtain 

[V]P-[VT]P=0     (modp). 

Finally,  the  existence  of  an  invariant  subgroup  HP} 
containing  T,  is  proved  as  in  §  66. 

Another  useful  theorem  based  on  the  same  principle  as  the  last 
two  is  the  following :  //  a  primitive  group  G  in  n  variables  contains  an 
abelian  subgroup  of  order  pa<J>  and  variety  k,  where  p  is  a  prime,  and 
a=/c,  then  G  contains  an  invariant  subgroup  Hp  which  does  not  make 
up  the  whole  of  G,  and  p  must  be  a  factor  of  n.  Cf .  reference  given  in 
Remark,  §  67. 

EXERCISES 

1.  Prove  that  no  primitive  group  in  four  variables  can  contain 
two  commutative  transformations  as  follows:    one  of  order  p  =  5, 
7,  11,  or  13;  the  other  of  order  q,  prime  to  p,  and  of  variety  4. 

2.  Under  what  conditions  may  two  commutative  transforma 
tions  of  different  prime  orders  p  and  q  belong  to  a  group  which  con 
tains  no  invariant  subgroups  H p  or  Hq  ? 

3.  Let  a  group  G  in  four  variables  contain  an  abelian  subgroup 
generated  by  the  group  E,  S-2  =  (l,  1,  -1,  -1),  &  =  (!,  -1,  -1,  1), 
<S4=(1,  —1,  1,  —1),  and  the  transformation  T  =  (l,  1,  w,  w2),  where 
w3  =  l.     Prove  that  G  contains  an  invariant  subgroup  of  order  3  , 
involving  T. 

69.  Unit-circle.  In  the  complex  plane,  a  root  of  unity 
a.  =  x+iy  =  cos  0+i  sin  0  represents  a  point  on  the  unit- 
circle  x2+y2=l.  Accordingly,  the  multipliers  of  a  trans 
formation  $=(ai,  02,  .  .  ,  an)  will  represent,  m  points 
on  the  unit-circle  if  S  is  of  variety  m.  Thus,  the  multi 
pliers  of  S  =  (i,  —  i)  represent  two  such  points,  separated 
by  an  arc  of  180°. 


ADVANCED  THEORY  OF  LINEAR  GROUPS         95 

We  find  it  convenient  in  this  and  the  next  few  para 
graphs  to  indicate  the  matrix  of  a  transformation  S  =  [ast] 
by  its  first  row  :  S  =  [an  a^  .  .  .  .  ain]  . 

LEMMA  1.  Let  S=(a1,  .  .  ,  an)  be  a  transformation 
in  canonical  form,  whose  multipliers  are  distributed  over  an 
arc  A  of  the  unit-circle  of  less  than  18GP,  and  let  T  be  a 
unitary  transformation  (§  19).  Then  no  element  in  the 
principal  diagonal  of  the  matrix  of  the  transformation 
V=  TST~l  can  vanish. 

Proof.  —  There  being  no  loss  of  generality  by  limiting 
ourselves  to  the  case  of  three  variables,  we  put  T  = 
[an  #12  #13].  Then  (§  19)  T~1  =  [du  021  031],  and  we  find 
TST-  1  =  [6n  612  6i3],  where 

^S  =  a,iasla1+a,2a6.2a2+a,3as3a3  («=1,  2,  3). 

Denote  the  number  astast  (which  is  real  and  positive 
unless  as,  =  0)  by  pt.  Then 


Now,  if  we  regard  a.1}  a^,  a3  as  unit  vectors  radiating 
from  the  center  of  the  unit-circle  to  points  of  the  arc  A, 
the  sum  bss  will  evidently  be  a  non-vanishing  vector 
lying  in  the  angle  subtended  by  the  arc  A.  Hence  the 
lemma. 

It  is  furthermore  evident  that  the  length  of  the  vector 
bss  is  less  than  the  sum  of  the  lengths  of  the  vectors  pi^, 
^2^2,  ??3a3  (namely  ^1+^2+^3  =  1),  unless  those  of  the 
vectors  just  mentioned  which  do  not  vanish  have  the  same 
direction.  Hence  the 

LEMMA  2.  No  element  bss  of  the  matrix  of  V  of 
Lemma  1  can  be  a  root  of  unity  (or  even  be  of  unit  length} 
unless  those  of  the  vectors  p\&\,  .  .  .  ,  Pnan  which  do  not 
vanish,  have  the  same  direction.  Hence,  if  bss  is  a  root  of 
unity,  and  if  neither  ast  nor  asv  vanish,  then  at  =  a,. 


96  FINITE  COLLINEATION  GROUPS 

70.  Theorem  8.  //  a  group  G  contains  a  trans 
formation  S=(a1}  .  .  .  ,  an)  whose  multipliers,  when 
located  on  the  unit-circle,  occupy  an  arc  ^=120°  and  extend 
ing  not  more  than  60°  on  either  side  of  some  one  of  them,  say 
aij  *  then  G  is  not  primitive.  (It  is  assumed  that  S  is  not  a 
similarity-transformation.) 

Let  n  =  3  to  begin  with.  We  assume  that  G  has  the 
unitary  form  and  that  at  the  same  time  S  has  the  canonical 
form  (Corollary,  §  22).  Furthermore,  we  assume  for 
the  present  that 

(11)  Ctl4=tt2,       al=Fa3, 


leaving  to  the  end  of  §  71  the  treatment  of  the  general 
case. 

We  shall  prove  in  order 

(A)  The  transformation  S  and  all  its  conjugates  in 
G  generate  an  intransitive  group  G'  (§71). 

(B)  Due  to  the  invariance  of  G'  (Exercise  3,  §  31), 
the  group  G  cannot  be  primitive  (§  72). 

71.  Proof  of  (A).  —  Since  the  multipliers  alt  a2,  a3  all 
lie  on  an  arc  ^120°,  no  element  in  the  principal  diagonal 
of  any  conjugate  of  S  can  vanish  (Lemma  1,  §69). 
Hence,  if  V=[bn  612  613]  is  such  a  conjugate,  then  &n=|=0. 
Now,  if  612  =  613  =  0  for  every  conjugate,  the  group  G' 
generated  by  these  conjugates  is  reducible  and  therefore 
intransitive  (§  20). 

The  proposition  (A)  is  therefore  established  by  proving 
612  =  613  =  0.  To  this  purpose  let  us  denote  by  M  the 
aggregate  of  all  those  conjugates  of  S,  if  any,  for  which 
at  least  one  of  the  two  elements  612,  bu  does  not  vanish. 

*  For  an  illustration,  take  the  transformation  of  order  6  in  three 
variables  whose  multipliers  are  al  =1,  a2  =  —  w,  a3  =  —  u>»,  where  w»=l; 
or  one  of  order  7  whose  multipliers  are  al  =  1,  aa  =cos  v  -f  i  sin  v,  a3  =  a2  —  i  = 
cos  v  —i  sin  B,  where  v  =(360/7)°. 


ADVANCED  THEORY  OF  LINEAR  GROUPS         97 

Let  T'  =  [a/!  a{2  a^]  be  any  one  of  these.     We  then  introduce 
a  f unction  f(T')  denned  as  follows: 


We  have 

(12) 

since  ai  ai  +a^  a/2  +a^  d^  =  1  (if  a^  a^  =  1,  it  would  of 
necessity  follow  that  a{2  =  a{a  =  0,  contrary  to  the  hypoth 
esis  concerning  M). 

We  now  let  T  =  [ast]  be  that  transformation  in  the 
aggregate  M  for  which  f(T)  has  the  greatest  value.  Con 
structing  Ti  =  TST~l  we  shall  then  demonstrate  the 
following  properties  of  T\: 

(1°)  /(7\)  >f(T) , *  (2°)  T,  belongs  to  M , 

thus  violating  the  assumption  as  to  T.  The  conclusion  is 
drawn  that  M  does  not  exist. 

To  prove  (1°),  put  02/04  =  cos  vz-\-i  sin  v2,  a3/a1  =  cos  vz 
+i  sin  vs,  and  we  find 


where 

cos  t>2+ai3«i3  cos 


Now,  by  the  condition  of  the  theorem, 

s  =  2,  3),  so  that  cos  vs 


*  The  use  of  an  inequality  of  this  kind  is  in  substance  a  discovery  of 
Valentiner's  (De  endelige  Transformations-Gruppers  Theori,  Copenhagen, 
1889,  p.  115).  An  equivalent  inequality  is  used  in  an  ingenious  manner 
by  Bieberbach  to  find  a  limit  to  the  orders  of  linear  groups.  (See 
Sitzungsberichte  der  Kgl.  Preussischen  Akademie  der  Wissenschaften,  1911, 
pp.  231-40;  also  two  papers  by  Probenius  in  the  same  Proceedings  (1911) 
pp.  241-48  and  pp.  373-78. 


98  FINITE  COLLINEATION  GROUPS 

Hence,  we  derive 

\)=A^  aiidn + i  (ai2«i2 + «i3«i3)  =  «iian + i  ( 1  —  «nan) 


Accordingly,  since  1+X2>2A  unless  A  =  1, 
by  (12),  it  follows  that 


To  prove  (2°),  we  first  notice  that  T^TST'1  is  a 
conjugate  of  S.  It  remains  for  us  to  show  that,  if  TI  = 
[6n  612  613],  612  and  613  do  not  both  vanish. 

Assume  the  contrary  :  612  =  613  =  0.  Then  bu&u  =  1, 
so  that  the  vector  bn  is  of  unit  length  (in  fact,  6n  must  be 
a  root  of  unity).  But, 


and  therefore,  since  an=}=0  (Lemma  1,  §  69),  and  since 
by  assumption  either  0,12  or  ctia^O,  say  a^^O,  we  have 
ai=a2  (Lemma  2,  §  69),  contrary  to  the  hypothesis  (11). 
Hence  TI  belongs  to  M  if  T  does. 

In  the  general  case,  the  multipliers  of  S  may  be  repeated. 
Assume  that  ah  a2  .    .    .  occur  respectively  ra,  k,   .    .    .   ,  times: 

S  =  (a.i,    .    .    .    ,  aij    a2,    .    .    .    ,  a2;    .    .    .    )  . 

The  aggregate  M  consists  here  of  all  those  conjugates  V  =  [bst] 
of  S  for  which  the  equations 

(13)     bst  =  0     (s  =  l,  2,   .    .    .   ,  ra;  t  =  m  +  l,  ra+2,   .    .    .   ,  n) 

are  not  all  satisfied:  As  above,  the  proof  of  (A)  depends  on  show 
ing  that  such  an  aggregate  does  not  exist;  and  to  this  end  the  func 
tion  }(T')  is  introduced,  which  now  has  the  form 


f(T')  =  +     (o{i  +aij+   .    .    .    +a'mm)(ai\ 

In  place  of   (12)  we  here  have  f(T')<m.     The    inequality   (1°) 
*(Ti)>f(T)  is  proved  without  difficulty. 


ADVANCED  THEORY  OF  LINEAR  GROUPS         99 

To  prove  (2°)  we  assume  first  that  Ti  =  TST~l  does  not  belong 
to  M;  that  is,  if  Ti  =  [bst],  we  assume  the  set  (13)  true.  Now  we 
write  TI  in  canonical  form  as  far  as  the  first  m  variables  are  con 
cerned;  S  will  still  have  the  canonical  form,  and  G  can  still  be  written 
as  a  unitary  group.  Moreover,  the  new  aggregate  M  will  coincide 
with  the  old.  If  then  TI  =  (0lf  02,  .  .  ,  Om)  as  far  as  the  first  m 
variables  are  concerned,  we  have 

m+i-f-    .    . 


dm 


By  using  the  Lemma  2,  §  69,  and  the  facts  that  T  =  [ast\  belongs  to 
M  and  that  none  of  the  numbers  an,  a22,  .  .  ,  amm  vanish,  we  prove 
readily  that  at  least  one  of  the  system  of  equations  just  written  is 
impossible.  Hence  T\  belongs  to  M. 

72.  Proof  of  (B);  general  case.  By  (A),  the  first  m 
variables  x\,  .  .  ,  xm  are  transformed  into  linear  func 
tions  of  themselves  by  S  and  its  conjugates,  if  ai  is  repeated 
m  times  among  the  multipliers  of  S.  The  group  Gf 
generated  by  these  conjugates  is  therefore  reducible,  and 
hence  also  intransitive;  say  in  the  two  sets  X=(xi, 
.  .  ,  xm),  Y=(xm+i,  .  .  ,  xn)  to  begin  with. 

Now  let  Gf  be  broken  up  into  its  ultimate  sets  of  in- 
transitivity  in  any  way  whatsoever,  without  reference  to 
the  original  intransitive  set  X.  Then  if  y\,  .  .  ,  yk  is  a 
set  of  intransitivity,  and  if  S  is  written  in  canonical  form 
as  far  as  this  set  is  concerned,  its  multipliers  within  the 
set  are  all  =ait  or  none  of  them  are.  (That  is,  in  the  first 
case  the  variables  are  linear  functions  of  x\,  .  .  ,  xm;  in 
the  last  case  they  are  linear  functions  of  £m+i,  .  .  ,  xn.) 
This  is  easily  seen  when  y\y  .  .  ,  yk  are  expressed  as 
functions  of  x\,  .  .  ,  xn,  and  the  condition  imposed  that 
S  transforms  these  functions  into  constant  multiples  of 
themselves.  Thus,  we  find  that  if  /  multipliers  are  equal 


100  FINITE  COLLINEATION  GROUPS 

to  ai,  and  the  remaining  multipliers  are  not,  then  /  vari 
ables,  say  2/1,  .  .  ,  2//,  are  functions  of  x\,  .  .  ,  xm  only, 
and  the  remaining  variables  2/7+1,  •  •  ,  Vk  functions  of 
xm+i,  .  .  ,  xn  only.  Since  these  two  sets  make  up  a 
single  intransitive  set  for  G',  and  the  sets  X,  Y  are  sepa 
rately  intransitive  sets,  it  follows  that  (2/1,  .  .  ,  2//)  and 
(2//+i,  .  .  ,  2//0  must  make  up  two  intransitive  sets. 

What  has  just  been  said  with  regard  to  S  will  be  true 
of  any  conjugate  of  S  within  G,  since  such  a  conjugate 
might  have  been  chosen  for  S  in  the  first  place  (conjugate 
transformations  have  the  same  multipliers;  cf.  Theorem 
11,  §  23).  Hence,  as  now  written,  G'  possesses  the  prop 
erty  that  any  one  of  the  conjugates  of  S  will  appear  partly 
in  the  canonical  form,  namely  in  regard  to  the  m  variables 
that  correspond  to  its  m  multipliers  ai. 

Now,  a  given  set  of  intransitivity  may  have  the  prop 
erty  that  several  conjugates  Si,  .  .  ,  Sp  have  the  canoni 
cal  form  (ai,  .  .  ,  ai)  for  this  set.  Calling  p  the  "  index" 
of  the  set  in  question,  we  seek  the  sets  of  highest  index  -*. 
Then  we  can  show  that  G  permutes  these  sets  among  them 
selves,  thus  proving  (B). 

Let  (zi,  .  .  ,  Zh)  be  such  a  set,  and  let  Si,  .  .  ,  S* 
be  the  corresponding  conjugates.  Selecting  any  trans 
formation  Fin  G,  we  write  V^SiV^  7\,  .  .  ,  V~1SnV  = 
Tn,  and  (zi)V  =  vi,  .  .  ,  (zh)V  =  vh.  Then  we  have, 


(zlt   .    .   ,  zh)SjV=(zi,   .    .   ,  zh)VTj        (j=l,  2, 
i.e.. 


The  variables  v\,  .  .  ,  vh  are  therefore  transformed  into 
<*-iVi}  .  .  ,  aiVh  by  the  maximum  number  of  distinct  con 
jugates,  namely  T\,  .  .  ,  Tn.  It  follows  that  v\,  .  .  ,  Vh 
are  linear  functions  of  the  variables  of  a  single  set  of  index 
TT.  For,  it  is  easily  seen  that  if  a  certain  linear  function 


ADVANCED  THEORY  OF  LINEAR  GROUPS        101 

v  of  the  variables  of  G  is  transformed  'into '<'£&'  by  tKo 
maximum  number  of  conjugates,  v  must,  of;  nectssity  be 
a  linear  function  of  the  variables  of  sorrte*  ontv  of  ttttf  sets 
of  index  v. 

73.  Theorem  9.  If  a  linear  group  G  contains  an 
abelian  subgroup  K  of  variety  m  (§  65)  and  order  k<f>,  where 
/^6m-i_(6m-2_|_6m-3_|_  +6) ,  then  G  is  intransitive 

or  imprimitive. 

We  can  prove  that  there  is  in  K  a  transformation  all  of 
whose  multipliers  are  not  more  than  ±60°  removed  from 
some  one  of  them  when  plotted  on  the  unit-circle. 
Theorem  8  may  then  be  applied. 

Let  K  be  written  in  canonical  form,  and  let  the  vari 
ables  be  arranged  in  m  sets  according  to  the  method 
explained  in  §  61.  The  case  n  =  ra  =  4  will  be  discussed  in 
detail;  it  will  then  be  sufficiently  obvious  how  to  prove 
the  theorem  for  the  general  case.  Furthermore,  since 
only  the  mutual  ratios  of  the  multipliers  are  of  importance 
in  Theorem  8,  a  slight  simplification  can  be  effected  by 
adopting  the  form  $'=(1,  /?/<*,  y/a,  •  •  )  instead  of 
S=  (a,  /?,  y,  .  .  ).  Thereby  we  gain  an  additional  advan 
tage:  the  order  of  the  group  composed  of  the  transfor 
mations  Sr,  .  .  is  k  when  the  order  of  the  corresponding 
group  S,  .  .  is  k*  (§  51,  3°). 

Accordingly,  we  have  fc^63  —  (62+6)  distinct  trans 
formations 

(14)        St  =  (l,at,pt,yt)  (f  =  l,2,   .    .   ,fc). 

We  now  plot  the  points  04,  a2,  .  .  ,  a^  on  the  unit-circle. 
Starting  at  one  of  these  points,  we  divide  the  circle  into 
6  equal  parts.  It  is  evident  that  one  at  least  of  these 
parts  will  contain  more  than  k/Q  points,  either  within 
or  upon  its  boundary.  (In  the  case  under  discussion, 
one  part  will  contain  at  least  62  — 6  points.)  There  are 


102  FINITE  COLLINEATION  GROUPS 

therefore  G2—  £  Dr  more  points  which  are  not  separated  one 
frotn  another  by  more  than  60°.  Let  the  corresponding 
iransforniatio'ris  of  the  set  (14)  be  denoted  by  Si,  Sz, 
.  .  ,  /Sao. 

We  next  plot  the  points  ft,  /32,  .  .  ,  fto  on  the  unit- 
circle.  Dividing  this  circle  as  before,  we  find  at  least 
30/6+1  points  lying  on  one  of  the  arcs  of  60°,  correspond 
ing  to  the  transformations  Si,  Sz,  .  .  $6.  We  finally 
plot  the  points  71,  .  .  ,  ye  and  find  at  least  2  points  on 
an  arc  of  60°.  If  the  corresponding  transformations 
are  Si,  Sz,  it  is  plain  that  their  corresponding  multipliers 
are  never  more  than  60°  apart.  It  follows  that  the  trans 
formation  $2$T1=(1,  at/a-i,  ft/A,  72/71)  fulfils  the  con 
ditions  imposed  on  S  in  Theorem  8,  the  root  ax  there 
being  unity  here. 

NOTE.  —  By  modifying  to  some  extent  the  principle  developed 
in  §§  69-72,  and  by  using  certain  processes  in  the  Geometry  of 
Numbers  (cf.  Transactions  of  the  American  Mathematical  Society, 
XV  (1914),  227),  the  author  has  obtained  a  very  much  lower  limit 
than  the  one  given  in  Theorem  9;  namely  /c^A^-1,  where  h  is  a 
number  lying  between  3  and  5,  depending  on  the  nature  of  the  prime 
factors  involved  in  the  order  of  the  group  K.  The  proof  of  this 
result  has  not  yet  been  published. 

EXERCISES 

1.  In  a  group  G,  a  transformation  all  of  whose  multipliers  lie 
on  an  arc  <  60°  is  commutative  with  one  whose  multipliers  lie  on  an 
arc  <  180°  (Frobenius). 

2.  Prove  that  if  a  group  G  contains  a  transformation  T  all 
of  whose  multipliers   lie  on  an  arc  <  72°,  then  G  is  not  primitive. 
(Hint:  Either  Tor  T5  fulfils  the  conditions  imposed  on  S  of  Theorem  8.) 

3.  By  Theorem  9,  a  primitive  group  G  cannot  contain  an  abelian 
subgroup  of  variety  3,  whose  order  k<f>  is  equal  to  or  greater  than  300. 
Examine  in  detail  the  possibilities  fc  =  26,  27,  28,  29,  and  prove  that 


74.  Order  of  a  primitive  group  in  n  variables.    We  are 
now  in  a  position  to  set  a  superior  limit  to  the  order  g<t> 


ADVANCED  THEORY  OF  LINEAR  GROUPS       103 

of  a  primitive  group  G  in  n  variables.  A  Sylow  subgroup 
of  order  pa  is  monomial  and  must  contain  an  abelian  sub 
group  of  order  pa~b  at  least,  where  pb  denotes  the  highest 
power  of  p  which  divides  n\  (Exercise,  §  61).  Hence, 
by  §  73,  Note, 


and  that  part  of  g  which  is  made  up  of  prime  factors  not 
greater  than  n  is  not  greater  than  w!5(n~1)fl(nty,  where 
0(ri)  denotes  the  number  of  primes  smaller  than  n+1. 
Again,  if  n-f-1  is  a  prime  and  is  a  factor  of  g,  the  cor 
responding  Sylow  subgroup  is  abelian  (Corollary,  §  61) 
and  its  order  is  ^5n-1.  Finally,  the  remaining  factor 
of  g  is  the  order  of  an  abelian  group  (§  62)  and  is  also 
^5n-1.  It  follows  that  the  order  of  a  primitive  group  in 
n  variables  is 


HISTORICAL  NOTE.  —  In  1878  Jordan  proved  the  classi 
cal  theorem  that  the  order  of  a  finite  linear  group  in  n 
variables  is  of  the  form  A/,  where/  is  the  order  of  an  abelian 
self-conjugate  subgroup,  and  where  A.  is  inferior  to  a  fixed 
number  which  depends  only  upon  n.*  Definite  limits 
to  A  have  been  given  by  Schur  for  the  case  where  the 
characteristics  belong  to  a  given  algebraic  domain  ;f  by 
Bieberbach  and  Frobeniust  and  by  the  author.  § 

Concerning  the  theorems  of  §§  60-68,  see  Transactions 
of  the  American  Mathematical  Society,  IV,  387-97;  V, 
310-20;  VI,  230-32;  VII,  523-29;  XII,  39-42. 

*  Journal  fUr  die  reine  und  angewandte  Mathematik,  Bd.  84,  p.  91. 

t  Sitzungsberichte  der  Kdniglich-Preussischen  Akadcmie  der  Wissen- 
schaften,  1905,  pp.  77  fl. 

t  Sitzungsberichte,  etc.,  1911,  pp.  231  ff.,  241  flf. 

§  Transactions  of  the  American  Mathematical  Society,  V  (1904),  320-21 
for  primitive  groups;  VI  (1905),  232  for  imprimitivo  groups. 


CHAPTER  V 
THE  LINEAR  GROUPS  IN  THREE  VARIABLES 

75.  Introduction.     The  determination  of  the  linear 
groups  in  three  variables  is  here  based  on  the  following 
classification  : 

1.  Intransitive  and  imprimitive  groups. 

2.  Primitive  groups  having  invariant  intransitive  or 
imprimitive  subgroups. 

3.  Primitive  groups  whose  corresponding  collineation 
groups  are  simple  ("  primitive  simple  groups"). 

4.  Primitive  groups  having  invariant  primitive  sub 
groups. 

No  specific  theory  is  needed  for  the  determination  of 
the  groups  in  the  first  class,  beyond  that  given  in  §  60. 
For  the  determination  of  the  groups  in  class  3  the  theorems 
of  §§  61-73  are  very  useful. 

The  notation  and  conventions  laid  down  in  §  51  are 
observed  throughout  the  chapter. 

76.  Intransitive  and  imprimitive  groups.    There  are 
two  types  of  intransitive  groups: 

(A)  Xi=ax{,  xz  =  fix!>,  x3  =  yx^         (abelian  type). 

(B)  xi  = 


i 

In  (B)  the  variables  z2,  x3  are  transformed  by  a  linear 
group  in  two  variables  (cf.  chap.  iii). 

The  imprimitive  groups  are  all  monomial.     There  are 
two  types: 

104 


LINEAR  GROUPS  IN  THREE  VARIABLES          105 

(C)  A  group  generated  by  an  abelian  group 

H:  Xi  =  ax[,     X2  =  Px?>,     Z3  =  y4 

and  a  transformation  which  permutes  the  vari 
ables  in  the  order  (xiX2x3).  This  transforma 
tion  may  be  written  in  the  form 

T:  xi  =  x?,,     X2  =  x't,     X3  =  x[, 

by  a  suitable  choice  of  variables. 

(D)  A  group  generated  by  H,  T  of  (C)  and  the  trans 

formation 

R:  x\  =  ax[,     #2  =  6x3,     xs  =  cxi. 

77.  Remarks  on  the  invariants  of  the  groups  (C)  and 
(D).  Interpreting  xi,  xz,  xs  as  homogeneous  co-ordinates 
of  the  plane,  the  triangle  whose  sides  are  Xi  =  Q,  #2  =  0, 
z3  =  0  is  transformed  into  itself  by  the  operators  of  (C) 
and  (D)  ;  in  other  words,  XiX^x^  is  an  invariant  of  these 
groups. 

Later  on  it  becomes  necessary  for  us  to  know  under 
what  conditions  there  are  other  invariant  triangles. 
Assuming  the  existence  of  one  such,  say 

(  1  ) 


we  operate  successively  by  the  transformations  of  H  and 
by  T.  Observing  that  a,  /?,  y  cannot  all  be  equal  for 
every  transformation  of  H  ,  as  otherwise  (C)  and  (D)  would 
be  intransitive,  we  find  by  examining  the  various  possi 
bilities  that  (1)  could  not  be  distinct  from  XiX2^3  =  0  unless 
H  is  the  particular  group  generated  by  the  transformations 

&  =  (!,  w,  <o2),      82  =  (w,  o>,  w)  ;     u>3=l. 


106  FINITE  COLLINEAT1ON  GROUPS 

There    are    then    four    invariant    triangles    for    (C), 
namely  : 


=  0; 

(2) 

(0=1,     CO,    Or    0)2). 

In  the  case  of  (D),  these  four  triangles  will  be  invariant 
if  the  group  is  generated  by  Si,  S2,  T  and  R,  the  latter  now 
having  the  form 


R:  x1=-x[ 


, 


either  directly  or  after  multiplication  by  suitable  powers 
of  Si  and  S2. 

78.  Groups  having  invariant  intransitive  subgroups. 
A  group  having  an  invariant  subgroup  of  type  (A)  is 
intransitive  or  imprimitive  (Lemma,  §  61),  and  a  group 
having  an  invariant  subgroup  of  type  (B)  is  intransitive.  ' 
For,  let  V  be  any  transformation  of  such  a  group,  and  T 
any  transformation  of  (B).  Then  VTV~1=Ti  belongs 
to  (B),  and  if  we  put  (xi)Ti  =  ax1}  (xi)V  =  y,  we  have 


This  shows  that  y  =  Q  is  an  invariant  straight  line  for  (B). 
But  xi  =  0  is  the  only  such  line,  and  therefore  y  =  (xi)  V  =  kx\, 
k  =  constant.  The  group  in  question  is  therefore  reduc 
ible  and  hence  intransitive. 

79.  Primitive  groups  having  invariant  imprimitive 
subgroups.  We  now  consider  a  group  G  containing 
an  invariant  subgroup  of  type  (C)  or  (D),  §  76.  These 
types  leave  invariant  the  triangle  Xix2x3  =  0  (§77),  and 
if  this  is  the  only  one,  we  could  prove  by  the  method  of 
§  78  that  G  would  also  transform  this  triangle  into  itself. 
But  then  G  would  not  be  primitive.  We  therefore  assume 


LINEAR  GROUPS  IN  THREE  VARIABLES          107 

that  there  are  four  invariant  triangles  for  (C)  and  (D), 
permuted  among  themselves  by  the  transformations 
of  G.  Let  us  denote  the  triangles  by  h,  k,  t3,  U,  in  order 
as  they  are  listed  in  (2). 

We  now  associate  with  each  transformation  in  G  a 
substitution  on  the  letters  ti,  k,  t3,  h,  indicating  the  manner 
in  which  the  transformation  permutes  the  corresponding 
triangles.  We  thus  obtain  a  substitution  group  K  on 
four  letters  to  which  G  is  multiply  isomorphic  (§32), 
and  the  invariant  subgroup  (C)  or  (D)  corresponds  to 
the  identity  of  K.  No  one  of  the  four  letters  could  be 
left  unchanged  by  every  substitution  of  K.  For  the 
corresponding  triangle  would  be  invariant  under  G;  and 
bringing  this  triangle  into  the  form  XiXzX3  =  0  by  a  suitable 
choice  of  new  variables,  G  would  appear  in  intransitive  or 
imprimitive  form.  Moreover,  no  transformation  can 
interchange  two  of  the  triangles  and  leave  the  other  two 
fixed,  as  may  be  verified  directly. 

Under  these  conditions  we  find  the  following  possible 
forms  for  K: 

(E')  1,  (MO  (W;* 

(F')  1,  (to)  (WO,  (MO  (MO,  (MO  (WO; 
(G')  the  alternating  group  on  four  letters,  generated 
by  (JifeXWO  and  (Ws«0. 


Now,  to  construct  the  corresponding  linear  transforma 
tions  we  observe  first  that  the  group  (D)  as  given  in  §  77 
contains  all  the  transformations  which  leave  invariant 
each  of  the  four  triangles.  Next  we  note  that  if  a  given 
transformation  V  permutes  the  triangles  in  a  certain 
manner,  then  any  transformation  V  which  permutes  them 
in  the  same  manner  can  be  written  in  the  form  V  =  XV, 
X  being  a  transformation  of  (D).  For,  V'V~l  must  leave 

*  The  throo  types  of  G  corresponding  to  the  three  substitution  groups  : 
E,  (/!*,)  (*3f4)  ;  K,  (^j)  (<2I4)  ;  E,  (V4)(V3>  aro  equivalent  (§51). 


108  FINITE  COLLINEATION  GROUPS 

fixed  each  triangle,  and  is  therefore  a  transformation  X 
as  defined. 

We  are  now  in  a  position  to  construct  the  required 
groups.  By  direct  application  we  verify  that  the  trans 
formations  U,  V,  UVU~l: 

U:    Xi  = 

V:   xi  = 


UVU-1: 


permute  the  triangles  in  the  following  manner: 


Accordingly,  since  all  the  groups  required  contain  a 
transformation  corresponding  to  (fib)(J&))  every  such 
group  must  contain  a  transformation  XV,  X  belonging 
to  (D).  Hence,  if  G  contains  (D)  as  a  subgroup,  it  also 
contains  V.  If,  however,  (C)  were  a  subgroup  of  G, 
but  not  (D),  then  either  V  is  contained  in  G,  or  else  XV, 
where  X  is  a  transformation  contained  in  (D)  but  not  in 
(C).  In  this  event  X  may  be  written  X\R,  where  X\ 
belongs  to  (C).  Hence,  finally,  either  V  or  RV  belongs 
to  G.  Howpver,  V2=(RV)2  =  R.  Thus  R,  and  therefore 
also  V,  are  contained  in  G  in  any  case. 

Again,  if  G  contains  a  transformation  corresponding 
to  (£2^4)  or  (t\t^(tzU),  such  a  transformation  can  be  written 
XU  or  XUVU-1,  X  belonging  to  (D).  Hence,  since  G 
contains  (D)  as  we  have  just  seen,  it  will  contain  either 
U  or  UVU'1  in  the  cases  considered.  We  therefore  have 
the  following  types: 


LINEAR  GROUPS  IN  THREE  VARIABLES          109 

(E)  Group  of  order  36<£  generated  by  (C)  as  given 
in  §77: 


and  the  transformation  V  of  (3). 

(F)  Group  of  order  72<£  generated  by  Si,  T,  V  and 

UVU~l. 

(G)  Group  of  order  216<£  generated  by  &,  T,  V  and  [7. 

These  groups  are  all  primitive,  and  they  all  contain 
(D)  as  an  invariant  subgroup.  The  group  (G)  is  called 
the  Hessian  group.* 

80.  Primitive  simple  groups  :f  the  Sylow  subgroups. 
In  order  to  utilize  Theorems  4-7,  chap.  IV,  it  becomes 
necessary  for  us  to  study  the  effect  of  the  presence  in  a 
group  G  of  an  invariant  subgroup  Hp;  or,  since  G  is  here 
to  be  a  simple  group,  \  to  determine  the  possibility  G  =  HP. 
Now,  it  is  at  once  seen  that  no  transformation 
T=  (ai,  a2,  a3)  whose  order  is  prime  to  p,  can  belong  to  Hp, 
by  the  results  of  §  67.  Accordingly,  the  order  of  Hp  is  a 
power  of  p.  However,  a  group  of  this  order  is  not  primi 
tive  (§61),  and  therefore  we  shall  dismiss  from  considera 
tion  in  §§  80-81  all  groups  which  may  be  shown  to  contain 
Hp,  as  for  instance  a  group  containing  the  transformation 
T=(—ljiti),  where  i  =  V  —  1  (cf.  §  66),  or  the  transforma 
tion  Tr=(o>i,  cw2,  «2w3),  where  a>5  =  co:j  =  G>jj=l,  and  where 
e  is  a  primitive  9th  root  of  unity.  On  the  other  hand, 
the  presence  of  the  transformation  T=(e<»i,  eo>2,  eo>3)  does 
not  imply  the  presence  of  a  subgroup  Hp. 

*  Cf.  Jordan,  Journal  fur  die  reine  und  angewandte  Mathematik,  Bd. 
84  (1878),  p.  209. 

t  Such  a  group  may  contain  the  group  of  similarity-transformations, 
which  is  an  invariant  subgroup  (cf.  §  51,  1°). 

t  If  G  contains  the  group  of  similarity-transformations,  we  can  assume 
that  //p  does  so  too. 


110  FINITE  COLLINEATION  GROUPS 

We  now  proceed  to  enumerate  the  different  possible 
types  of  Sylow  subgroups.  In  this  list,  the  symbol  P& 
means  "  Sylow  subgroup  of  order  ft." 

p  =  2: 

(a)  P2,  generated  by  (1,  -1,  -1). 

(b)  P4,  generated  by  (1,  -1,  -1) 

and  (-1,   1,  -1). 

(c)  P4,  generated  by  (1,  i,  —i). 

(d)  P8,  generated  by  (6)  and  Tf: 


(e)  PS,  generated  by  (c)  and  T'. 


(a)  P3,    generated  by  (1,  w,  o>2)  . 

(b)  P3,j>,  generated  by  (e,  c,  ew2),  where  c3  =  o>. 

(c)  Pg,*,,  generated  by  (a)  and  (6). 

(d)  PW,  generated  by  (a)  and  T":   x\  =  xj,  x2  =  zj, 
x3  =  x[. 

(e)  Pzit,  generated  by  (6)  and  T". 

A  group  of  order  5a  is  abelian  and  contains  no  trans 
formation  of  order  52  (§  67,  Corollary).  If  therefore 
a  =2,  we  necessarily  have  two  or  more  generating  trans 
formations,  say  $i  =  (ai,  a2,  a3)  and  >S2=(a4,  a5,  a6),  both 
of  order  5.  If  now  a  represents  a  primitive  5th  root  of 
unity,  a  transformation  of  order  5  and  variety  2  can  be 
found  in  G,  namely  T  =  SfS*  =  (a,  a,  a~2).  (We  merely 
determine  a  and  b  such  that  a«a^  =  a^a|  .) 

The  transformation  T  leaves  invariant  every  straight 
line  through  the  point  X:  Xi  =  x2  =  0.  If  R  be  any  con 
jugate  to  T,  then  R  would  likewise  leave  invariant  every 
straight  line  through  a  certain  point  Y.  Accordingly, 
both  T  and  R  leave  invariant  the  straight  line  which  joins 


LINEAR  GROUPS  IN  THREE  VARIABLES          111 

X  and  Y.  Let  the  variables  be  so  changed  that  this  line 
is  £i  =  0;  the  group  generated  by  T  and  R  will  then  be 
intransitive  (§  20).  If  the  component  H  of  this  group 
in  the  variables  x%,  x3  is  not  primitive,  T  and  R  are  com 
mutative,  because  they  will  appear  in  the  canonical  form. 
Therefore,  either,  1°  all  the  transformations  conjugate 
with  T  are  mutually  commutative,  or,  2°  for  at  least 
one  pair  of  such  transformations  (say  T  and  R)  there  is 
a  primitive  group  H  in  two  variables  (x2,  x3).  This 
group  must  be  of  type  (E),  §  58,  and  contains  the  follow 
ing  transformation  (  —  w,  —  w2).  Therefore  G  contains 
the  transformation  (1,  —  o>,  —  w2),  and  is  not  primitive 
(§  70). 

In  the  alternate  case  1°,  G  contains  invariantly  the 
abelian  group  generated  by  T  and  its  conjugates  and  is 
not  primitive  (§61). 

Accordingly,  a=l,  and  a  transformation  of  order  5 
must  be  of  variety  3.  By  trial,  we  now  find  the  following 
type: 

P5,  generated  by  (1,  a,  a  :),  ;-5—  1 . 

A  Sylow  subgroup  of  order  7a  is  abelian.  By  §  73, 
a<2,  and  by  §70,  we  can  have  no  transformation  of 
type  (a2,  a2,  a3)  or  (1,  a,  a~!),  where  a7=l.  Accordingly, 
we  are  limited  to  the  following  type: 

P7,  generated  by  (ft  /82,  /?<);  (?=  1 . 

The  types  P5,  PI  are  mutually  exclusive.  For,  if 
both  were  present  in  a  group,  we  should  have  a  transforma 
tion  of  order  35  (§  90,  Exercise  2),  and  therefore  two 
commutative  transformations  of  orders  5  and  7,  namely 
those  listed  above  under  P6  and  P7.  But  this  would 
imply  a  subgroup  Hp  (§68). 


112  FINITE  COLLINEATION  GROUPS 

For  the  same  reasons,  the  types  (6),  (c),  and  (e)  for 
p  =  3  cannot  exist  simultaneously  with  a  subgroup  of  order 
5  or  one  of  order  7  in  a  primitive  group. 

81.  The  orders  of  the  primitive  simple  groups.    Con 
sider  first  the  case  of  a  group  G  of  order  g<t>  =  lg\<t>.    By 
§80,  gi  is  a  factor  of  23 . 32</>,  and  we  have  8  or  36  subgroups 
of  order  7  (§  36)  in  the  corresponding  collineation  group. 
In  the  first  case,  P7  is  invariant  in  a  group  H'  of  order 
h'  =  g<f>/8  (§  30),  and  in  the  second  case  it  is  invariant  in  a 
group  of  order  0<£/36.     Now,  PI  is  generated  by  a  trans 
formation  (ft,  /32,  /34)  as  we  have  seen;    it  is  therefore  a 
simple  matter  to  show  that  the  groups  of  orders  63<£  or 
14<£  would  be  impossible  or  would  imply  a  subgroup  Hp. 
It  follows  that  0  =  252  or  168. 

Consider  next  a  group  G  of  order  g<£  =  5</i</>,  g\  being 
as  above  a  factor  of  23.32.  Here  we  have  6  or  36  Sylow 
subgroups  of  order  5.  Now,  it  is  easily  verified  that  such 
a  group,  generated  by  (1,  a.  -~-),  cannot  be  invariant 
in  a  subgroup  H'  of  ovlor  h'  =  2Qk  or  15fc;  it  follows  that 
the  ni^bcrs  gr/6  and  gr/36  are  either  5  or  10;  i.e.,  g  is  one 
or  the  numbers  30,  60,  180,  360. 

The  orders  to  be  considered  are  therefore  60<£,  360<£ 
and  168<£  (cf.  §  48).  There  are  no  simple  groups  of  order 
2a.36  (§100). 

82.  The  three  types  of  primitive  simple  groups.    The 
simple  groups  of  orders  60  and  360  are  simply  isomorphic 
with  the  alternating  groups  on  5  and  6  letters  respectively. 
To  determine  the  corresponding  linear  groups,  we  con 
struct  the  transformations  F\,  F2,   .    .   ,  satisfying  the 
relations  specified  in  §  50,  as  was  done  in  chap.  Ill  in  the 
case  of  the  group  (E).     We  find  the  types:* 

*  Of.  Maschke.  Mathcmatische  Annalen,  Bd.  51  (1899),  pp.  264-67. 


LINEAR  GROUPS  IN  THREE  VARIABLES          113 

(H)  Group  of  order  60,  generated  by  F\,  F2,  F3,  satis 
fying  the  relations: 

p«  =  FI=FI  =  E,  (F1F2Y 
namely  : 

Fi:  X!  =  x!>,  x2  =  x^,  x3  = 
F2  =  (l,  -1,  -1); 


where     ni  =  %(-l  +  l/S),   /*2  =  J(-l-l/5)  . 
(I)    Group  of  order  360<£,  generated  by  Fi,  F2,  F3  of 
(H)  and  F±,  where 


F4:  xi=-x,    x2=-o>X3,    x3= 

Concerning  the  simple  group  of  order  168,  the  generat 
ing  relations  can  best  be  obtained  by  examining  the  sub 
stitution  group  on  7  letters  simply  isomorphic  with  the 
group  in  question.  *  We  shall  here  select  the  substitutions 
S=(abcdefg),  T=(adb)(cef),  R=(ab)(ce)\  satisfying  the 
relations  : 


(J)    Group  of  order  168  generated  by  S,  T,  R: 


T:  Xi  =  x2,  x2  =  X3,  x3  =  x[ 
R:  Xi  = 


where    £*  =  !,  a  =  ^34-/33,  b  =  P2-p,  c  =  p-pG; 


*  Cf.  Burnsido,  Theory  of  Groups  of  Finite  Order,  2d  ed.,  Cambridge, 
1911,  pp.  218,  310;  Miller,  Blichfeldt,  and  Dickson,  Theory  and  Applica 
tions  of  Finite  Groups,  New  York,  1916,  p.  50. 


114  FINITE  COLLINEATION  GROUPS 

83.  Primitive  groups  having  invariant  primitive  sub 
groups.  A  possible  subgroup  Hp  is  monomial  §  (80). 
We  have  already  determined  the  primitive  groups  con 
taining  a  group  of  this  type  invariantly;  it  is  therefore 
unnecessary  to  consider  the  groups  containing  the  sub 
groups  Hp.  Hence,  the  orders  of  the  groups  still  to  be 
examined  must  be  factors  of  the  numbers  23  .  33<£,  23  .  32  .  5<£, 
23.32.7<£,  as  we  have  seen. 

The  group  (I)  has  already  a  maximum  order.  Again, 
the  group  (J)  cannot  be  invariant  in  a  group  of  order 
23.32.7<£,  since  a  possible  group  of  this  order  must  contain 
a  subgroup  Hp  (§  81).  As  to  the  group  (H),  we  make  use 
of  the  fact  that  its  6  subgroups  of  order  5  must  be  per 
muted  among  themselves  by  an  assumed  larger  group 
(K),  in  which  it  is  self-conjugate.  In  any  event,  we  would 
have  a  subgroup  of  order  20k  or  30k,  containing  a  given 
P5  self-conjugately.  But  this  would  imply  a  group  Hp 
(cf.  §  81). 

There  remain  the  groups  (E),  (F),  (G).  Now,  any 
larger  group  would  permute  among  themselves  the  four 
triangles  which  form  an  invariant  system  for  these  groups. 
But  this  was  just  the  condition  under  which  the  three 
given  groups  were  determined.  Hence  no  new  types 

result. 

EXERCISES 

1.  Determine  the  linear  groups  in  two  variables  by  the  method 
of  this  chapter.     (To  determine  the  primitive  simple  groups,  show 
first  that  the  order  of  such  a  group  is  a  factor  of  600.) 

2.  Obtain  the  group  (H)  by  the  second  method  given  in  §  58, 
in  the  following  form  : 

S'  =  (l,  e4,  e);     [/';    Xi=  —xit  x*=  -X3,  X3=  -X2; 

T':     3i  = 


where  eB  =  l,  s  =  e2-r-e3, 


LINEAR  GROUPS  IN  THREE  VARIABLES          115 

Show  also  that  the  second  form  of  (E),  §  58,  transforms  the 
variables  yo=  —  XiX2,  yi=xl,  yi=—x\  into  linear  functions  of  them 
selves,  and  hence  this  group  appears  as  a  linear  group  in  three 
variables,  which  is  precisely  the  group  (H)  as  given  in  this  exercise 
if  we  write  yo,  y\,  y3  for  x\t  x2,  x3  respectively  in  (H). 

3.  Obtain  the  group  (I)  by  adding  a  transformation  W  =  (ad)(ef) 
to  the  list  S'}  Uf,  T'  of  Exercise  2,  and  show  that 

W:  a;i  = 


where  X1  =  i(-  1=^^15),        *2  =  l(-l=r\/^l5)  . 

4.  Show  that  the  function  xlx3+x2Xi+xs3x2  is  invariant  under  (J). 

5.  In  §  80  it  was  shown  that  two  transformations,  both  of 
variety  2,  would  generate  an  intransitive  group.     By  the  same 
process  (geometrical),  prove  that  two  transformations  in  four  vari 
ables,  both  of  variety  2,  would  also  generate  an  intransitive  group. 

Bibliography.  —  Complete  discussions  of  the  linear 
groups  in  three  variables  are  to  be  found  in  the  following 
articles:  Jordan,  Journal  fur  die  reine  und  angewandte 
Mathematik,  Bd.  84  (1878),  pp.  125  ff.;  Valentiner,  De 
endelige  Transformations-Gruppers  Theori,  Copenhagen 
(Videnskabernes  Selskabs  Afhandlinger),  1889;  Mitchell, 
Transactions  of  the  American  Mathematical  Society,  XII 
(1911),  207-42;  and  the  author,  Transactions  of  the  Ameri 
can  Mathematical  Society,  V  (1904),  321-25,  and  Mathe- 
matische  Annalen,  Bd.  63  (1907),  pp.  552-72.  See  also 
Miller,  Blichfeldt,  and  Dickson,  Theory  and  Applications  of 
Finite  Groups,  §  125,  where  further  references  will  be  found. 


CHAPTER  VI 
THE  THEORY  OF  GROUP  CHARACTERISTICS 

84.  Introduction.  This  theory,  of  importance  not 
only  for  linear  groups,  but  for  abstract  and  substitution 
groups  as  well,  was  initiated  by  Frobenius  and  is  largely 
due  to  him,*  though  Schurj  and  BurnsideJ  simplified 
the  theory  and  added  extensive  applications.  We  shall 
devote  this  chapter  to  an  exposition  of  the  main  points 
of  the  theory  by  a  process  differing  in  many  respects  from 
previous  expositions. 

It  is  not  assumed  in  this  chapter  that  the  transforma 
tions  of  a  linear  group  under  discussion  are  necessarily 
of  determinant  unity. 

For  the  sake  of  uniformity  of  notation,  we  shall 
throughout  this  chapter  denote  the  order  of  a  group  under 
discussion  by  g  (unless  otherwise  specified),  however  the 
group  may  be  designated  (6r,  G',  H,  etc.)  Furthermore, 
the  number  of  its  sets  of  conjugate  operators  (§  29)  shall 
be  denoted  by  h,  these  sets  to  contain  respectively 
0i,  02,  .  .  ,  Qh  operators,  so  that  0  =  0i+02+  .  .  +gh- 

*  Sitzungsberichte  der  Kgl.-Preussischen  Akademie  der  Wissenschaften, 
1896,  pp.  985,  1343;  1897,  p.  994;  1899,  p.  482. 

t  Sitzungsberichte,  etc.,  1905,  p.  406;  Journal  fur  die  reine  und  ange- 
wandte  Mathematik,  Bd.  127  (1904),  pp.  20-50;  Bd.  132  (1907),  pp.  85-137; 
Bd.  139  (1911),  pp.  155-250. 

t  Ada  Mathematica,  XXVIII  (1904),  369-87;  Proceedings  of  the 
London  Mathematical  Society,  1904,  pp.  117-23;  Theory  of  Groups  of  Finite 
Order,  2d  ed.,  Cambridge,  1911,  pp.  243  ff. 

See  also  the  following  accounts:  Molien,  Sitzungsberichte,  etc.,  1897, 
pp.  1152-56;  Dlckson,  Annals  of  Mathematics,  1902,  pp.  25-49;  Miller, 
Blichfeldt,  and  Dickson,  Theory  and  Applications  of  Finite  Groups,  New 
York,  1916,  pp.  257-78;  and  the  author,  Transactions  of  the  American 
Mathematical  Society,  V  (1904),  461-66. 

116 


THE  THEORY  OF  GROUP  CHARACTERISTICS  117 

85.  Remarks  on  intransitive  groups. — Extension  of 
group-concept.  1°.  Let  G=(Si,  S2,  .  .  ,  Sg)  be  an 
intransitive  group,  which  upon  a  suitable  change  of  vari 
ables  breaks  up  into  two  or  more  groups,  G',  G",  .  .  , 
corresponding  to  the  various  sets  of  intransitivity.  We 
shall  have  occasion  to  employ  a  notation  analogous  to  that 
used  for  linear  transformations,  namely, 

(POO 

0     G"  0 
0     0     G'" 


and  we  shall  say  that  G',  G",   .    .     are  " component" 
groups  of  G. 

2°.  Under  certain  conditions  it  is  convenient  to  extend 
the  name  " group"  to  a  set  of  operators  which  are  not  all 
distinct,  but  which  can  be  put  into  a  (1,  1)  correspondence 
with  a  group  G.  A  group  G'  of  order  g'  which  is  (1,  h) 
isomorphic  with  G  can  be  exhibited  as  if  it  were  a  group 
simply  isomorphic  with  G,  namely  by  repeating  each  of 
its  operators  h  times.  For  instance,  the  substitution 
group  of  order  6:  E,  (ab),  (ac),  (be),  (abc),  (acb)  is  multiply 
isomorphic  with  two  of  its  subgroups:  E;  and  E,  (ab). 
With  the  concept  of  "group"  extended  as  indicated  above, 
we  may  exhibit  the  three  groups  as  simply  isomorphic  in 
the  following  manner: 

E,  (ab),   (ae),  (be),   (abc),   (acb); 
E,  E,       E,      E,      E,        E; 
E,  (ab),   (ab),  (ab),  E,        E. 

We  shall  accordingly  agree  to  look  upon  the  groups  G, 
G',  G",   .    .     in  1°  as  simply  isomorphic. 

86.  Characteristics.  1°.  Definition. — Let  the  vari 
ables  of  the  group  G  =  (Si,  82,  .  .  ,  Sg)  be  x\t  xz,  .  .  ,  xn. 


118  FINITE  COLLINEATION  GROUPS 

As  noted  in  §  23,  the  sum  of  the  multipliers  of  a 
given  transformation  St  is  called  the  characteristic  of  St, 
and  we  shall  here  denote  it  by  xt  or  x(^)-  It  is  equal 
to  the  sum  of  the  elements  in  the  principal  diagonal  of 
St;  if  St  is  written  in  canonical  form:  (a,  /3,  .  .  ',  A.), 


2°.  Inverse  and  conjugate-imaginary  transformations.  — 
The  multipliers  are  always  roots  of  unity,  and  if  a,  /?,  .  . 
are  the  multipliers  of  St,  those  of  S~l  are  the  reciprocals 
a"1,  /J-1,  .  .  .  Since  the  reciprocal  of  a  root  of  unity 
a  is  its  conjugate-imaginary  (a~1  =  a),  we  have 


3°.  Conjugate  transformations.  —  The  characteristics  of 
conjugate  transformations  are  equal  (§  23).  Hence,  if 
there  are  h  complete  conjugate  sets  of  transformations  in 
G  (§  29),  there  cannot  be  more  than  h  different  charac 
teristics  of  G.  In  conformity  with  the  notation  adopted  in 
§  84,  we  shall  indicate  these  by  the  symbols  xi,  X2>  •  •  >  Xh> 

4°.  Intransitive  groups.  —  The  characteristic  of  a  trans 
formation  S  of  the  intransitive  group  G,  §  85,  1°,  is  evi 
dently  the  sum  of  the  characteristics  of  the  ''component" 
transformations  S',  >S",  .  .  : 

x(S)=x(S')+x(S")+   .   .  . 

5°.  Substitution  groups.  —  When  a  substitution  con 
sisting  of  a  single  cycle  on  more  than  one  letter, 
S=(aiOz  .  .  .  am),  is  written  in  matrix  form  as  a  linear 
transformation  (§1),  the  elements  in  its  principal  diagonal 
are  all  zero.  Hence  its  characteristic  is  zero.  If  w=l, 
the  transformation  has  the  form  S=(l);  in  this  case 
\(S)  =  l.  The  multipliers  of  *S=(aia2  .  .  .  am)  are  the 
m  different  rath  roots  of  unity;  thus,  the  substitution 
$  =  (0110203),  when  written  as  a  linear  transformation  in 
canonical  form,  becomes  S=(l,  <*>,  <*)2),  where  o>3  =  l. 


THE  THEORY  OF  GROUP  CHARACTERISTICS  '119 

It  follows  that  the  characteristic  of  the  most  general 
substitution  S  on  n  letters  is  the  integer  which  equals  the 
number  of  cycles  of  one  letter;  i.e.,  \(S)=the  number  of 
letters  that  the  substitution  leaves  unchanged.  In 
particular,  the  characteristics  of  a  regular  group  (§  47) 
are  all  zero,  except  that  \(E)  equals  the  order  of  the  group. 

Frobenius  and  Schur  call  the  set  of  quantities  Xi,  •  •  ,  Xg 
a  character  of  G.  In  their  terminology,  an  abstract  group  G  would 
possess  as  many  simple  characters  as  there  are  non-equivalent  linear 
groups  to  which  G  is  simply  or  multiply  isornorphic  (cf.  §  99). 

87.  The  sum  and  product  of  matrices.  The  sum  of  a 
series  of  square  matrices  of  the  same  order  is  the  matrix 
whose  elements  are  the  algebraic  sums  of  the  correspond 
ing  elements  of  the  given  matrices.  Thus, 

62 


a\     61]      [02 
ci     di.rLca 


If  Si,  Sz,  .  .  are  linear  transformations  in  the  same 
variables,  we  shall  write  &+&+  .  .  to  denote  the 
matrix  which  is  the  sum  of  the  matrices  of  Si,  $2,  .  •  • 

The  product  of  two  matrices  is  obtained  by  the  rule 
given  in  §  3,  irrespective  of  whether  the  matrices  represent 
linear  transformations  or  not.  If  M  represents  a  matrix, 
and  c  a  constant  or  a  variable,  the  symbol  cM  shall  repre 
sent  the  matrix  obtained  by  multiplying  every  element 
of  M  by  c. 

In  accordance  with  these  definitions  we  find 


Si  +  $2  = 

.    .  )  = 

.    )T=T-1SiT+T-1S*T+   .    .  , 


c(Si+S2+   .    .  )= 


120  FINITE  COLLINEATION  GROUPS 

Again,  if  X  represents  a  linear  function  of  the  variables 
xi,  xzt  .  .  ,  xn,  and  if  the  matrix  M  be  regarded  as  the 
matrix  of  a  linear  transformation  (though  the  determinant 
of  M  may  vanish),  we  shall  by  (X)M  represent  the  result 
of  operating  upon  X  by  M  (cf.  §  2).  Now  if 


where  Si,   .    .   ,  Sm  are  linear  transformations  of  a  group 
in  the  variables  x\,   .    .   ,  xnj  it  is  then  readily  seen  that 

(X)M=(X)Sl+(X)&+   .    .   +(X)Sm. 

88.  Invariants.  A  function  /(xi,  .  .  ,  xn)  which  is 
transformed  into  a  constant  multiple  of  itself  by  every 
transformation  of  a  group  G  is  called  an  invariant  of  G. 
It  is  an  absolute  invariant  if  the  constant  multiplier  is 
unity  for  every  transformation  of  G;  otherwise  it  is  a 
relative  invariant. 

A  series  of  invariants  /i,  .  .  ,  fm  are  said  to  be  inde 
pendent  of  each  other  if  the  variables  cannot  be  eliminated 
from  the  equations 


where  a\,  .  .  ,  am  are  arbitrary  constants.  They  are 
said  to  be  linearly  independent  if  no  identity  exists  of  the 
form 

61/1+    •    •    - 


where  61,   .    .   ,  bm  are  constants,  not  all  zero. 

EXERCISES 

1.  The  substitution  group  E,  (xixd(x&t),  (x#$(x&d,(x 

is  abelian  and  consequently  intransitive  when  written  as  a 
linear  group  (§  22).  Write  it  in  canonical  form,  and  show  that  it 
possesses  four  invariants  of  the  first  degree,  one  of  which  is  absolute. 
Show  also  that  the  characteristics  are  4,  0,  0,  0. 

2.  Write  the  symmetric  group  on  three  letters  as  a  linear 
group,  and  construct  the  matrices  which  are  each  the  sum  of  the 


THE  THEORY  OF  GROUP  CHARACTERISTICS     121 

transformations  of  a  conjugate  set.  Then  show  that  any  one  of 
these  matrices  (M)  and  any  transformation  (S)  of  the  group  are 
commutative:  MS  =SM.  Show  also  that  the  matrices  are  mutually 
commutative. 

3.  Write  down  the  characteristics  of  the  group  in  the  previous 
exercise,  and  prove  that  their  sum,  the  sum  of  their  squares,  the 
sum  of  their  cubes,  etc.,  is  always  a  multiple  of  6,  the  order  of  the 
group. 

4.  Construct  the  two  independent  invariants  of  the  first  degree 
of  the  group  of  order  6  in  §  95. 

Prove  also  that  the  alternating  group  on  5  letters  possesses 
only  one  invariant  of  the  first  degree. 

5.  Prove  that  the  characteristics  of  the  two  transformations 
AB  and  BA  are  equal. 

ON   THE   CHARACTERISTICS   OF   TRANSITIVE   GROUPS, 

§§89-91 

89.  Theorem  1.  //  S\,  .  .  ,  Sm  are  the  different 
transformations  of  a  conjugate  set  of  a  transitive  linear  group 
G  in  n  variables,  then  the  matrix 

M  =  S,+   ...+&, 

is  commutative  with  every  transformation  of  G  and  has  the 
form  of  a  similarity-transformation  (a,  a,  .  .  ,  a),  where 
a  =  mx(Si)/n. 

Proof. — That  M  is  commutative  with  any  given  trans 
formation  T  of  G  is  seen  as  follows.  We  have 

T-1MT=T-lSiT+T-1S*T+    .    .    =Si+S2+   .    .    =  M, 

since  T~1S\T,  .  .  ,  T~lSmT  are  the  transformations 
Si,  .  .  ,  Sm  over  again  in  some  order  (§  29,  (c)).  Hence 
MT=TM. 

We  shall  proceed  to  prove  the  second  part  of  the 
theorem.  By  the  method  of  §  21,  we  can  find  a  linear 
function  of  the  variables  of  the  group  which  is  an  invariant 
of  M,  say  (xi)  =axi,  where  a  is  a  constant  (possibly  zero). 


122  FINITE  COLLINEATION  GROUPS 

There  may   be  other  linear  functions  xz,   .    .   such  that 


let  xi,  x2,  .  .  ,  xk  be  all  those  that  have  this  property 
and  are  linearly  independent.  We  shall  indicate  this  fact 
symbolically  by  (xi}  .  .  ,  xk)M  =  a(xi,  .  .  ,  xk).  It  is 
plain  that  a  linear  function  having  this  property  must 
be  a  linear  function  of  the  variables  x\,  .  .  ,  xk. 
Now  let  T  be  any  transformation  of  G,  and  let 

y2,    .     .    ,    (xk)T  =  yk. 


Then,  since  (zi,   .    .   ,  xk)MT=(xi,   .    .   ,  £&)  TAT,  we  have 


so  that  2/1,  .  .  have  the  property  ascribed  above  to 
xi,  .  .  .  It  follows  that  2/1,  .  .  are  linear  functions  of 
xi,  .  .  ;  that  is,  the  latter  variables  form  an  intransi 
tive  set  of  G  (§20),  unless  k  =  n.  Hence,  since  G  is 
transitive,  k  =  n,  and  M  has  the  form  of  a  similarity- 
transformation  (a,  a,  .  .  ,  a). 

Finally,  to  find  the  value  of  a,  we  observe  from  the 
formation  of  M  that  the  sum  of  the  elements  of  its  princi 
pal  diagonal,  ri«,  equals  the  sum  of  the  characteristics  of 
Si,  .  .  ,  Sm.  But  these  are  all  equal  (§86,  3°);  hence 


90.  Theorem  2.  Let  the  number  of  transformations 
in  the  h  different  conjugate  sets  of  a  transitive  group  G  in  n 
variables  be  g\,  gz,  .  .  ,  Qh,  and  let  the  corresponding 
characteristics  be  denoted  by  \\,  X2,  .  -  ,  Xh  (§  86,  3°). 
Then 


where  c8tv,  .    .    .  represent  certain  positive  integers  or  zero. 


THE  THEORY  OF  GROUP  CHARACTERISTICS     123 

Proof.  —  Let  the  sum  of  the  matrices  of  the  gk  trans 
formations  of  the  kth  set  be  denoted  by  Mk.     Then 


8 

(8,     «=1,      2,     .     .     ,h). 

For  the  gsgt  matrices  in  the  product  M8Mt  must  make 
up  one  or  more  conjugate  sets,  since  T~l(MsMt)T  = 
(T-1M8T)(T-lMtT)=MsMt.  Accordingly,  this  prod 
uct  is  the  sum  of  one  or  more  of  the  matrices  Mi,  .  .  ,  Mh, 
possibly  repeated  a  certain  number  of  times. 

We  now  substitute  in  (2)  the  canonical  forms  of  the 
matrices  Mi,  .  .  ,  Mh  as  given  by  Theorem  1,  and  obtain 
the  equation  (/3,  .  .  ,  0)  =  (y,  .  .  y,  ),  where  ft  has  for 
value  the  left-hand  member  of  (1),  and  y  the  right-hand 
member. 

COROLLARY.  //  a  transitive  linear  group  G  in  n 
variables  contains  two  characteristics  Xs>  Xt  such  that  the 
sum  of  the  n2  roots  in  the  product  XsXt  cannot  be  written  as  a 
sum  in  which  primitive  roots  of  index  k  are  absent,  then 
there  is  in  G  a  characteristic  containing  roots  of  index  k 
and  therefore  a  transformation  whose  order  is  k  or  a 
multiple  of  k* 

This  follows  from  the  equation  (1).  By  the  condi 
tions  of  the  corollary,  at  least  one  of  the  characteristics 
Xv  of  the  right-hand  member  must  contain  roots  of  index 
k.  There  is,  therefore,  a  transformation  whose  order  is 
divisible  by  k.  For,  the  order  m  of  a  transformation 
S  =  (a,  /3,  .  .  )  is  the  least  common  multiple  of  the  indices 
of  the  roots  a,  /?,  .  .  ,  since  Sm  =  (l,  !,..)  = 


To    illustrate,    let    x«= 
where  i  —  V—\  and  a  is  a  primitive  fifth  root  of  unity. 
Here  \sXt)  or  4ia  -\-2ia?  —  2a—  a3,  cannot  be  written   as  a 


*Burnside,  Theory  of  Groups,  2d  ed.,  p.  347. 


124  FINITE  COLLINEATION  GROUPS 

sum  which  is  free  from  roots  of  index  20  (namely  ia,  ia2, 
etc.)  by  Kronecker's  theorem  (§  133,  6°). 

EXERCISES 

1.  Selecting  the  h  equations  (1)  obtained  by  keeping  s  fixed 
while  taking  t  =  l,  2,   .    .   ,  h,  prove  that  gsXs/n  is  an  algebraic 
integer  (Frobenius). 

2.  Prove  that  if  a  group  in  n  variables  contains  transformations 
of  orders  p  and  q,  two  different  prime  numbers  both  greater  than 
n-f  1,  then  the  group  contains  a  transformation  of  order  pq. 

91.  Theorem  3.  The  sum  of  the  characteristics  of  a 
transitive  group  in  n  variables,  n==^2,  is  zero. 

Proof. — Let,  as  in  §  90,  the  group  G  contain  h  conju 
gate  sets,  and  let  Mk  represent  the  sum  of  matrices  in  the 
fcth  set.  By  §  89,  the  matrices  M\,  .  .  ,  Mh  all  have  the 
form  of  a  similarity-transformation;  the  same  will  there 
fore  be  the  case  with  the  sum  M  of  all  the  matrices  in  G, 
which  is  Mi+M2+  .  .  -\-Mh,  say  M=(t,  e,  .  .  ,  e). 
Now,  if  S  is  any  transformation  of  G  distinct  from  the 
identity,  the  relation  MS  =  M  (§  27,  Exercise  3)  is  found 
to  imply  e  =  0.  Hence, 

01x1+02x2+  .  .  +ghXh  =  Q- 

When  w=l,  two  cases  arise.  If  the  group  consists 
of  E  repeated  g  times  (§  85,  2°),  the  sum  in  question  is  g; 
if  the  transformations  are  not  all  the  identity,  the  sum 
vanishes.  For  an  illustration  take  the  group  of  order 
4m  (each  of  the  following  transformations  repeated  m 
times):  E=  (1),  S=  «,  Sz  =  (-1),  S*  =  (-*);  t-V^-1. 

Combining  these  results  and  referring  to  §  86,  4°, 
and  §  85,  1°,  we  get  the 

COROLLARY.  The  sum  of  the  characteristics  of  an 
intransitive  group  G  is  its  order  multiplied  by  the  integer 
representing  the  total  number  of  the  component  groups 


THE  THEORY  OF  GROUP  CHARACTERISTICS     125 

in  G  whose  transformations  are  the  identity  repeated  g 
times. 

It  is  plain  that  to  each  group  in  G  of  the  latter  type 
we  have  an  absolute  invariant  of  the  first  degree,  and  vice 
versa.  Hence  we  may  state  the  preceding  corollary 
in  the  following  form:  the  sum  of  the  characteristics  of  a 
group  G  is  its  order  multiplied  by  the  total  number  of  inde 
pendent  absolute  invariants  of  G  of  the  first  degree. 

EXERCISE 

Prove  that  the  average  number  of  letters  which  remain  un 
changed  by  a  substitution  of  a  transitive  substitution  group  G  is 
equal  to  unity.  (Hint:  Prove  that  G  possesses  a  single  absolute 
invariant  of  the  first  degree.) 

ON   THE    CHARACTERISTICS   OF   ISOMORPHIC   GROUPS, 
§§  92-94 

92.  Composition  of  two  groups.     Let  G'  and  G"  be 

two  simply  isomorphic  groups  in  respectively  n  and  m 
variables,  say  x\,  .  .  ,  xnj  and  y\,  .  .  ,  ym.  Then  the 
nm  products  x\y\,  .  .  ,  xnym  are  transformed  into 
linear  functions  of  themselves  when  operated  upon  simul 
taneously  by  two  corresponding  transformations  S'  and 
S".  Hence,  regarding  these  combined  transformations 
as  a  single  linear  transformation  S  in  the  nm  variables, 
we  obtain  a  linear  group  G  simply  isomorphic  with  G'  and 
G".  For,  to  E  of  G'  and  G"  will  correspond  E  of  G,  and 
if  SiSi  =  S'r,  S'JS'j  =  S'r',  we  have  8^  =  Sr.  We  shall  say 
that  G  is  compounded  from  the  groups  G'  and  G". 

LEMMA  1.  The  characteristic  of  a  transformation  S 
contained  in  a  group  G  which  is  compounded  from  the  groups 
G'  and  G",  is  equal  to  the  product  of  the  characteristics  of 
the  corresponding  transformations  S'  and  S". 

This  is  seen  immediately  when  S'  and  S"  are  both 
written  in  the  canonical  form,  which  can  obviously  be 


126  FINITE  COLLINEATION  GROUPS 

done.      If    then     (xa)Sf  =  aaxa,     (yb)S"  '  =  PbVb,    we    find 
=  a.apb(xayb),  so  that 


0  =  1     6  = 

LEMMA  2.     Let  there  be  given  an  invariant  of  G: 
f=XlYl+   .    .   +XkYk, 

where  X\,  .  .  ,  Xk  are  linear  functions  of  Xi,  .  .  ,  xn 
and  Yi,  .  .  ,  Yk  linear  functions  of  y\,  .  .  ,  ym.  Then 
if  k<n,  the  group  G'  is  intransitive. 

Proof.  —  For  the  sake  of  simplicity  take  fc  =  3,  say 
f=XiYi+X2Y2+XtYt.  We  may  assume  that  Xi,  X2,  X* 
(as  well  as  Y\,  F2,  F3)  are  linearly  independent  of  each 
other;  if  it  were  possible  to  write,  say,  Xs^ai 
the  function  /  could  be  expressed  in  two  terms  : 


A  transformation  of  G'  will  change  X\,  X*,  X3  into 
there  linearly  independent  functions  of  Xi,  .  .  ,  xn,  and 
the  corresponding  transformations  of  G"  will  change 
Yi,  F2,  F3  into  three  linearly  independent  functions  of 
t/i,  .  .  ,  ym.  Let  the  resulting  expression  be 


and  we  should  have  /=/'.  But  this  implies  that  X{, 
X^  X'3  are  linear  functions  of  X\,  X%,  X*.  Hence,  if  n>3, 
G'  is  reducible  and  accordingly  intransitive. 

93.  Theorem  4.  Let  a  transitive  group  G  be  com 
pounded  with  its  conjugate-imaginary  group  G.  The  re 
sulting  group  is  intransitive,  and  among  the  component 
groups  into  which  it  breaks  up  will  be  found  just  one  group 


THE  THEORY  OF  GROUP  CHARACTERISTICS     127 

made  up  of  the  identity  repeated  g  times.     Hence  (Corol 
lary,  §  91), 

(3)  01x1x1+02x2x2+  .  -  +OhXkXh  =  g- 

Proof.  —  Let  the  compounded  group  be  denoted  by  H. 
Its  characteristics  are  \tXt  (Lemma  1,  §92);  and  their 
sum  2  (i.e.,  the  left-hand  member  of  (3)),  divided  by  g, 
represents  the  number  of  absolute  invariants  of  H  of  the 
form  XiXi+  .  .  +XnXn  (Corollary,  §  91)  ;  or,  as  we  may 
write  it  by  a  proper  distribution  of  the  terms, 

.    +Xnxn. 


We  know  one  such  invariant  already,  namely  the 
Hermitian  invariant  (§  18),  and  we  may  assume  that  the 
variables  are  originally  so  chosen  that  this  invariant  is 


.   .  -rxnxn. 
Then  if  X  is  any  constant,  the  expression 

/-|-X/=  (Xi-{-hXi)Xi-\-(X2-\-kXz)X2-\-     .      .     -j-(Jfn-}-X:rn):rn 

is  also  an  invariant. 

Now,  the  constant  X  may  always  be  determined  such 
that  Xi+^Xi,  .  .  ,  Xn-\-Xxn  are  not  linearly  independent. 
(For  an  illustration,  take  n  =  2,  X\  =  pxi+qxz, 

Here  X  is  a  solution  of  the  equation 


q 


=  0.) 


r        s+X 

Therefore  either  G'  is  intransitive  (Lemma  2,  §  92), 
or  /+A.7  vanishes  identically.  Hence,  since  the  first 
alternative  violates  the  assumption  of  the  theorem,  any 
invariant/  of  H  is  merely  a  constant  multiple  of  /  (viz., 
/=— X/);  in  other  words,  the  number  2/0  of  linearly 
independent  invariants  of  H  is  unity.  The  theorem 
follows  by  §  91. 

COROLLARY.     The  number  of  variables  n  of  a  transitive 
linear  group  G  is  a  factor  of  the  order  g. 


128  FINITE  COLLINEATION  GROUPS 

Proof.  —  Dividing  the  equation  (3)  by  n  we  get 


The  quantities  —  i  are  algebraic  integers  (Exercise  1, 

§  90),  as  well  as  the  quantities  \t-  Hence,  since  the  sums 
and  products  of  algebraic  integers  are  again  algebraic 
integers,  it  follows  that  g/n  is  an  algebraic  integer;  i.e., 
g/n  is  an  ordinary  integer  (§  134). 

94.  Theorem  5.  Let  Gr  and  G"  be  two  simply  iso- 
morphic  transitive  groups,  whose  corresponding  character 
istics  are  x't  o^nd  x't>  Then  the  sum 

(4) 


equals  g  or  zero,  according  as  the  two  groups  are  equivalent 
or  not. 

Proof.  —  If  they  are  equivalent,  the  variables  of  G" 
may  be  chosen  so  that  the  groups  are  identical,  and  the 
conjugate-imaginary  groups  G'  and  G"  will  be  identical 
also.  The  sum  (4)  is  then  equal  to  g,  by  Theorem  4. 

Conversely,  if  the  sum  equals  g,  the  two  groups  are 
equivalent,  as  we  shall  proceed  to  prove.  Let  the  vari 
ables  of  G'  and  G"  be  respectively  Xi,  .  .  ,  xn;  y\,  .  .  ,  ym; 
and  suppose  that  n^m.  The  expression  (4),  divided 
by  g,  equals  the  number  of  absolute  invariants  of  the 
form  X\Yi+XzYz-\-  .  .  .  of  the  group  G  compounded 
from  G'  and  G"  (§91).  Since  there  is  one  such,  n  —  m 
(Lemma  2,  §  92),  and  we  may  choose  the  variables  of  G" 
so  that  the  invariant  has  the  form 

I  =  xiyi+'xzy2+    .    .    +xnyn. 

But  this  is  the  form  of  the  invariant  of  the  group 
H  compounded  from  G'  and  its  conjugate-imaginary 
group  G',  after  the  variables  of  the  latter  group  have  been 


THE  THEORY  OF  GROUP  CHARACTERISTICS     129 

written  3/1,  ..,  yn  in  place  of  x\,  .  .  ,  xn.  It  is  there 
fore  an  easy  matter  to  prove  that  the  groups  G"  and  G' 
are  identical.  For,  let  S',  S',  S"  be  corresponding  trans 
formations  of  G',  G',  G",  and  we  have  (I)S'S'=  (I)S"S't 
so  that  (I)S'=(I)_S."  (The  transformations  S'  and  S', 
as  well  as  S'  and  S",  are  commutative,  since  the  variables 
of  the  respective  groups  are  independent  of  each  other.) 
It  follows  directly  that  S'  and  S"  are  identical. 

Finally,  we  cannot  have  more  than  one  invariant  of 
the  form  /  above,  since  otherwise  the  groups  G'  and  G" 
would  be  intransitive  (cf.  proof  of  Theorem  4).  Accord 
ingly,  the  sum  (4)  is  either  g  or  zero. 

COROLLARY  1 .  //  the  corresponding  characteristics  of  two 
simply  isomorphic  groups  are  equal,  the  groups  are  equivalent. 

COROLLARY  2.  Let  x't  and  x't  be  corresponding  char 
acteristics  of  two  groups,  G'  and  G",  the  first  of  which  is 
transitive.  The  sum  (4),  divided  by  g,  will  in  this  case 
represent  the  number  of  components  in  the  group  G"  which 
are  equivalent  to  Gr,  when  the  former  is  broken  up  into  its 
ultimate  sets  of  intransitivity. 

We  write  G"  in  such  an  ultimate  intransitive  form 
(cf.  §  85,  1°),  and  apply  §  86,  4°,  and  Theorem  5. 

EXERCISES 

1.  Let  H'  and  H"  be  two  simply  isomorphic  intransitive  groups, 
and  let  their  corresponding  characteristics  be  denoted  by  x't  and 
x't-     Then  when  these  groups  are  split  up  into  component  transitive 
groups,  the  latter  may  not  all  be  non-equivalent.     Let  us  suppose 
that  GI,  G-2  ,   .    .   ,  Gn  represent   types  of  all  the  non-equivalent 
groups  obtained,  there  being  a:  and  61  of  the  first  type  in  H'  and  H" 
respectively,  a2  and  62  of  the  second  type,  etc. 

Now  prove  that  the  sum  (4),  §  94,  is  here  equal  to  the  integer 
0(ai&i+a2&2+  .  .  +anbn). 

2.  Prove  that  if  a  transitive  linear  group  of  order  g  in  n  vari 
ables   contains   a   subgroup   of   order  /  composed   of   similarity- 
transformations,  then  g  is  divisible  by  fn  (Schur).     (Hint:    Prove 


130 


FINITE  COLLINEATION  GROUPS 


first  that  if  xt  does  not  vanish,  there  will  be/  distinct  sets  of  conju 
gate  transformations  for  which  the  products  gtxtxt  in  (3),  §  93,  have 
the  same  value.) 

3.  Prove  that  a  group  G  containing  a  subgroup  P  of  order  p, 
a  prime  number  ^n2/(n  —  1),  is  intransitive,  unless  P  is  invariant 
in  a  larger  subgroup  of  G.  (Hint:  With  reference  to  the  sum  (3), 
prove  that  the  sum  of  the  terms  xtxt  corresponding  to  the  transfor 
mations  of  order  p  in  P  alone  is  ^np—ri2,  by  Exercise  1  above. 
Hence  prove  that  the  sum  of  such  terms  from  all  the  subgroups 
conjugate  to  P  and  from  the  identity  exceeds  g.} 

ON   THE   TOTALITY   OF   NON-EQUIVALENT   ISOMORPHIC 
GROUPS,    §§95-99 

95.  The  regular  group.  We  shall  now  consider  the 
regular  substitution  group  H  (§  47).  Regarded  as  a 
linear  group,  it  is  intransitive,  since  it  possesses  an  abso 
lute  invariant  of  the  first  degree,  namely  the  sum  of  the 
letters  of  substitution. 

As  an  illustration,  take  the  regular  group  on  the  letters 
Xi,  .  .  ,  x6,  simply  isomorphic  with  the  symmetric 
group  on  three  letters.  Its  substitutions  are  as  follows: 

Si  =  E,  £4 


83  = 


If  we  now  introduce  new  variables  zi,   .    .   ,  26,  where 


the  transformations  of  H  will  all  be  of  the  following  type: 

"a  0  0  0  0  0" 
060000 
0  0  c  d  0  0 
0  0  e  f  0  0 
0  0  0  0  c  d 
L.O  0  0  0  e  f_ 


(5) 


THE  THEORY  OF  GROUP  CHARACTERISTICS  131 

showing  that  H  is  split  up  into  4  component  transitive 
groups  in  respectively  1,  1,  2,  2  variables.  The  groups 
in  the  variables  23,  z4  and  z5,  z6  are  equivalent,  both  being 

generated  by  the  transformations  &  =  (<*>2,  *)»&"" It  AT 

Hence,  there  are  three  non-equivalent  groups. 

In  general,  a  regular  group  H  on  g  letters,  when  written 
as  a  linear  group  in  g  variables,  is  intransitive.  Cor 
responding  to  its  ultimate  sets  of  intransitivity  we  have  a 
number  of  component  transitive  linear  groups,  which  are 
not  all  non-equivalent.  If  we  select  a  representative  of 
each  set  of  equivalent  groups,  we  get,  say,  k  representa 
tives,  forming  a  set  of  non-equivalent  component  groups  of 
H.  Let  them  be  denoted  by  H',  H",  .  .  ,  #<*>,  and  their 
corresponding  characteristics  by  \t,  x't,  •  •  ,  X^- 
Hence,  if  H  contains  n'  groups  equivalent  to  H', 
n"  group  equivalent  to  H",  .  .  ,  the  characteristic  of 
the  transformation  St  of  H  is  (§  86,  4°) 

(6)  x($)=n'xl+n"x7  +    -    -    +n<»x(f>, 

and  this  sum  is  =g  or  =0,  according  as  St  =  E  or  St^E 
(§  86,  5°). 

96.  Theorem  6.  Let  G  be  a  transitive  linear  group  in 
n  variables,  and  let  H  be  a  regular  substitution  group  on  g 
letters  simply  isomorphic  with  G.  Then  among  the  com 
ponent  transitive  groups  into  which  H  breaks  up,  there  are 
just  n  groups  equivalent  to  G. 

The  theorem  follows  from  Corollary  2,  §  94,  if  we  let 
G  and  H  represent  Gf  and  G"  respectively.  The  sum  (4) 
will  here  be  equal  to  ng  (cf.  §  95). 

The  number  k  of  non-equivalent  component  groups 
of  H  is  equal  to  h  (§  99).  Anticipating  this  result,  we 
can  now  say  that  the  number  of  non-equivalent  transitive 
linear  groups  to  which  a  given  transitive  linear  group  G 


132  FINITE  COLLINEATION  GROUPS 

is  simply  or  multiply  isomorphic  (cf.  §§  32,  85)  is  equal  to  the 
number  of  sets  of  conjugate  transformations  of  G. 

EXERCISE 

Let  the  component  groups  H't  .  .  ,  #w  of  H  contain  respec 
tively  n't  n",  .  .  ,  n(A)  variables.  Prove  that  (cf.  Exercise  1, 
§94) 


At  least  one  of  the  numbers  n',  .   .  is  unity.     If  there  is  more 
than  one,  then  H  possesses  a  relative  invariant  and  is  not  simple. 

97.  Theorem  7.     Given  any  two  transformations  S8 
and  St  of  H.     The  sum 


vanishes  if  Ss  is  not  conjugate  to  S^1;  if  these  two  trans- 
formations  both  belong  to  the  conjugate  set  denoted  by  the 
subscript  s,  then  3  =  g/gs. 

Proof.  —  Let  the  equation  (1),  §  90,  be  multiplied  by  n2, 
and  a  corresponding  equation  formed  for  every  one  of  the 
non-equivalent  groups  H',  .  .  ,  Hw  of  H.  The  con 
stants  cstv  are  the  same  for  each  of  these  groups,  since 
they  are  the  same  for  all  simply  isomorphic  groups. 
Hence,  adding  the  resulting  k  equations,  the  left-hand 
member  of  the  new  equation,  say  Fi  =  Fz,  will  be  gsg^, 
and  the  right-hand  member  the  expression 


k 

Now,  2^.n(w)xV  is  tne  sum  (6),  §  95,  for  the  subscript 

W=l 

v.     Hence,  this  sum  vanishes  or  is  equal  to  g,  according 
as  Sv  does  not  or  does  represent  the  identity.     Moreover, 


THE  THEORY  OF  GROUP  CHARACTERISTICS     133 

in  the  latter  case  gv  =  l.     Accordingly,   the  right-hand 
member  is  equal    to  gc^i,   under  the    assumption  that 


The  value  of  csti  is  found  from  (2),  §  90,  and  represents 
the  number  of  times  the  transformation  E  occurs  in  the 
product  MsMt.  Now,  let  R  be  a  transformation  of  Ms. 
If  R~l  is  found  in  Mt,  then  it  is  readily  seen  that  Mt  is  the 
sum  of  the  inverses  of  the  transformations  of  Ms;  in 
that  case  E  will  occur  gs  times  in  the  product  MsMt, 
and  gs  =  gt  =  cst\.  On  the  other  hand,  if  R~l  does  not 
occur  in  Mt,  neither  will  the  inverse  of  any  transforma 
tion  in  Ms;  in  this  case  csti  =  Q.  Interpreting  the  equa 
tion  Fi  =  F2  under  these  results  we  finally  prove  the 
theorem. 

The  second  alternative  in  Theorem  7  can  evidently  be 
stated  in  the  form  (cf  .  §  86,  2°)  : 

x/x/+xs"x/'+  .  .  +xs(k)xs(k)  =  g/ga. 

EXERCISES 

1.  Prove  that  the  set  of  non-equivalent  groups  simply  iso- 
morphic  with  the  alternating  group  on  5  letters  consists  of  the  group 
(H),  §  82;   the  group  (Hi)  obtained  by  interchanging  the  numbers 
Mi  and  /*2  in  (H)  ;  and  two  groups  in  4  and  5  variables  respectively. 

2.  Prove  that  the  group  (H),  §  82,  compounded  with  its  conjugate- 
imaginary,  splits  up  into  three  transitive  groups,  one  in  1  variable 
(the  identity  repeated  60  times,  corresponding  to  the  Hermitian 
invariant),  one  in  3  variables,  and  one  in  5  variables. 

98.  Group-matrix.  Let  x\,  .  .  ,  xn  be  the  variables 
of  a  transitive  or  intransitive  group  G=(Si,  .  .  ,  Sg), 
and  let  7/1,  .  .  ,  y0  be  g  variables,  independent  of  each 
other  and  of  xi,  .  .  ,  xn.  Then  the  matrix  (cf.  §  87) 


.   +ygSg 
is  called  the  group-matrix  of  the  group  G.     Thus,  the 


134 


FINITE  COLLINEATION  GROUPS 


group-matrix  of  the  regular  group  H  of  order  6  given  in 
§  95  has  the  form 


(7) 


M  = 


Vl  2/2  2/3  2/4  2/5  2/6 

2/3  2/1  2/2  2/5  2/6  2/4 

2/2  2/3  2/1  2/6  2/4  2/5 

2/4  2/5  2/6  2/1  2/2  2/3 

2/5  2/6  2/4  2/3  2/1  2/2 

.2/6  2/4  2/5  2/2  2/3  2/1. 


The  n2  elements  in  M  are  linear  homogeneous  func 
tions  of  y\j  .  .  ,  yg.  These  functions  may  not  all  be 
independent  of  each  other;  thus,  .there  are  only  6  inde 
pendent  elements  in  M  of  (7).  Bearing  in  mind  the 
definitions  of  §85,  2°,  we  shall  prove  the  following: 

THEOREM  8.  Let  H',  H",  .  .  ,  #<*>  form  a  com 
plete  set  of  non-equivalent,  simply  isomorphic,  transitive 
linear  groups  in  respectively  n',  n",  .  .  ,  nw  variables. 
Then  the  (n')2+(n")2+  .  .  +  (n^)2  =  g  elements  con 
tained  in  the  k  group-matrices  ?/i$i+  .  .  +2/A  of  these 
groups  are  all  independent  functions  of  the  variables 


Proof.  —  Consider  the  group-matrix  M  of  the  regular 
group  H  of  which  H',  H",  .  .  ,  H(k)  are  component 
non-equivalent  groups.  The  matrix  M  will,  while  H 
has  still  the  form  of  a  substitution  group,  contain  just  g 
independent  elements.  For,  each  element  will  consist 
of  a  single  letter  yj}  as  may  be  proved  easily  (cf.  §  47). 

Now  let  H  be  broken  up  into  its  different  component 
groups  by  means  of  a  linear  transformation  T  (cf.  §  13). 
Correspondingly  M  is  transformed  by  T  into  a  new  matrix 
M': 

T~lMT=T-l(ylSl+   .    .   +ygSa)T  = 


THE  THEORY  OF  GROUP  CHARACTERISTICS     135 

The  elements  of  M'  are  linear  functions  of  the  elements 
of  M,  and  vice  versa;  the  coefficients  being  functions  of 
the  elements  of  T.  Hence  there  are  as  many  independent 
functions  of  y\,  .  .  ,  yg  among  the  elements  of  M'  as 
among  the  elements  of  M;  namely  g. 

Now,  each  non-equivalent  component  group  H(i)  is  re 
peated  as  an  equivalent  group  n(j)  times  (§  96),  and  these 
equivalent  groups  can  be  made  identical  by  a  proper 
modification  of  T.  Correspondingly,  the  n(j-)  matrices 
involved  in  M'  will  have  the  same  elements.  Hence, 
there  will  be  at  most  as  many  independent  elements  in 
M'  as  are  found  in  a  set  of  non-equivalent  groups,  namely 
(n')2+(n")2_j_  _  _  +(w(*))2.  But  this  number  is  equal 
to  g  (Exercise,  §  96).  It  follows  that  all  these  elements 
are  independent,  and  the  theorem  is  proved. 

In  the  case  of  the  group  H  of  order  6,  the  matrix  M' 
will  have  the  form  (5),  §  95,  where  now 

a  =  2A  +  2/2  +2/3  +2/4  +2/5  +  2/6         &  =  2/i+2/2+2/3  —  2/4  —  2/5  —  2/6, 


/6>  /=2/i+a)2/2+w22/3- 

EXERCISES 

1.  Prove  that  the  n2  elements  of  each  of  the  matrices  of  the 
transformations  of  a  transitive  group  G  do  not  satisfy  a  linear 
homogeneous  equation,  whose  coefficients  are  the  same  for  every 
transformation  (Burnside). 

2.  Prove  that  if  a  certain  element  ouc  vanishes  in  every  trans 
formation  S  =  [aat]  of  a  group  G,  the  subscripts  u,  v  being  given,  then 
G  is  not  transitive  (Maschke). 

99.  Theorem  9.  The  number  k  of  non-equivalent 
transitive  linear  groups  into  which  the  regular  group  H  breaks 
up  (§  95)  is  equal  to  the  total  number  of  sets  of  conjugate 
substitutions  of  H.  In  other  words,  a  given  transitive 
linear  group  G  can  be  simply  or  multiply  isomorphic  with 
just  h  non-equivalent  transitive  linear  groups,  including 


136  FINITE  COLLINEATION  GROUPS 

itself  and  the  group  consisting  solely  of  the  transforma 
tion  E. 

The  proof  follows  that  of  Theorem  8  closely,  after  we 
have  first  made  equal  to  each  other  those  of  the  variables 
2/i,  .  .  .  ,  ya  which  are  factors  of  conjugate  transforma 
tions  in  the  matrix  M.  If  therefore  H  contains  h  con 
jugate  sets  of  respectively  g\,  .  .  .  ,  QH  transformations, 
we  shall  have  h  independent  variables,  say  1*1,  .  .  .  ,  vh. 

The  matrix  M'  now  has  the  form  of  a  transformation 
in  canonical  form.  Thus,  the  matrix  M'  in  the  group  of 
order  6  given  in  §  95  becomes 

M'=(a,  b,  c,  c,  c,  c), 

where  a  =  v\  +  2y2  +  8^3,  b  =  vi  +  2%  —  3%,  c  =  v\  —  v2.  In  fact, 
it  follows  by  Theorem  1,  §  89,  that,  as  far  as  the  variables 
of  H(j)  are  concerned,  M'  will  appear  in  the  form  of  a 
similarity-transformation  (ft,  ft,  .  .  .  ,  ft),  where 


If  H&  and  #(>'>  are  equivalent  groups,  ft=ft-/;   if 
and  #(/)  are  non-equivalent,  ft/=t=/2/  (cf.  §  94,  Corollary  1). 

Accordingly,  among  the  g  multipliers  of  M'  in  its 
new  form,  there  will  be  just  k  that  are  distinct,  and  these 
can  certainly  not  furnish  more  than  k  expressions  linearly 
independent  in  v\,  .  .  .  ,  v^  On  the  other  hand,  the 
matrix  M  will  contain  just  h  linearly  independent  elements, 
namely  Vi,  .  .  .  ,  tv  Hence  k^h  (cf.  proof  of  Theorem 
8),  and  h  of  the  multipliers  ft,  .  .  .  ,  ft  are  linearly 
independent,  say  ft,  ft,  .  .  .  ,  ft.  These  h  expressions 
can  therefore  not  all  vanish  unless  v\  =  v%  =  .  .  .  = 
*-0. 

However,  if  k>h,  the  expressions  ft,  .  .  .  ,  ft  must  all 
vanish  if  for  v\,  .  .  .  ,  Vh  we  put,  respectively,  the  conju- 
gate-imaginaries  of  the  characteristics  x!A+1)>  •  •  •  >  X(f+1) 
of  the  group  Gh+i-  (§  94).  But  these  quantities, 


THE  THEORY  OF  GROUP  CHARACTERISTICS     137 

Xih+l\  •  -  •  i  Xhh+l}  are  n°t  all  zero  (°nc  °f  them 
represents  the  number  of  variables  of  H(h+l)).  We 
conclude  that  k>h.  Hence,  finally,  k  =  h. 

AN   APPLICATION   OF   THE    PRECEDING   THEORY 

100.  Theorem  10.  No  simple  group  can  be  of  order 
paqb,  p  and  q  being  different  prime  numbers.* 

The  proof  is  divided  into  two  parts:  (A).  If  H  is 
the  regular  substitution  group  simply  isomorphic  with  a 
group  of  order  g  =  paqb,  assumed  simple,  then  one  of  the 
component  non-equivalent,  transitive  linear  groups  H', 
.  .  ,  Hw  contains  qa  variables,  and  one  of  the  conju 
gate  sets  of  H  contains  p?  transformations.  (B).  Under 
these  conditions  an  impossible  equation  is  derived. 

(A).  The  relation  0  =  py  =  (n')2+(n")2-h  .  ,  +(n<»)2 
with  the  conditions  that  the  numbers  nf,  .  .  .  ,  nw  are 
all  factors  of  g  (§  93,  Corollary)  and  that  only  one  of 
them  is  unity  (§  96,  Exercise),  implies  that  at  least  one 
of  them  is  greater  than  unity  and  prime  to  p;  say  n(t)  = 


Again,  the  relation  g  =  paqb  =  gl+  g2+  .  .  .  +& 
with  the  conditions  that  the  numbers  firi,  .  .  .  ,  fito  are 
factors  of  g  (§  29)  and  that  only  one  of  them  is  unity 
(or  there  would  be  an  invariant  operator),  implies  in  the 
same  manner  that  one  of  them  is  a  power  of  p;  say  gs  = 


(B).  We  now  have  a  transitive  group  Hw  in  n(t')=^1 
variables,  and  a  conjugate  set  of  gs  =  p?  transformations. 
Let  S(i)  denote  one  of  the  transformations  of  this  set, 
\(t)  its  characteristic,  and  T  the  corresponding  trans 
formation  (substitution)  of  H.  We  have  (§  90,  Exercise 
1,  and  §133,  7°): 


*  Burnside,  Proceedings  of  the  London  Mathematical  Society,  Series  2, 
I  (1904),  388-92. 


138  FINITE  COLLINEATION  GROUPS 

where  K  represents  the  sum  of  a  finite  number  of  roots  of 
unity.  It  follows  that  x(t)  =  QaKf)  K'  being  such  a  sum 
also  (I.e.).  But,  x(/)  is  already  the  sum  of  qa  roots  of 
unity.  Hence,  either  all  these  roots  are  alike,  or  x(f)  =  0*. 
The  first  supposition  makes  S(t)  a  similarity-transforma 
tion,  which  would  be  self-conjugate  in  Hw.  This  being 
impossible  for  a  simple  group,  we  infer  that  x(^=Q; 
and  this  not  only  for  the  group  H(t\  but  also  for  every  one 
of  the  groups  H',  .  .  ,  H(h)  (and  their  equivalent 
groups),  the  number  of  whose  variables,  like  n(t\  does 
not  contain  p  as  a  factor. 

Hence,  the  characteristic  of  T  in  H  is  the  expression 


when  account  is  taken  of  the  fact  that  one  of  the  numbers 
ft',  .  .  .  ,  n(h\  say  nf,  is  unity,  and  that  the  correspond 
ing  characteristic  x(1)  =  l  (cf.  §95).  Accordingly  (§86, 

5°), 

l+pK"-Q. 

But  such  an  equation  is  impossible  by  Kronecker's 
Theorem  (§  133,  6°).  We  conclude  that  H  is  not  simple. 

EXERCISE 

Prove  that  a  group  in  which  the  number  of  operators  in  a  con 
jugate  set  is  the  power  of  a  prime  number,  is  not  simple  (Burnside). 

*This  follows  directly  from  §  133,  6°. 


CHAPTER  VII 
THE  LINEAR  GROUPS  IN  FOUR  VARIABLES 

101.  Introduction.  We  shall  again  adopt  all  the 
conventions  laid  down  in  §  51,  and  we  shall  employ  the 
same  classification  for  the  groups  now  under  discussion  as 
that  used  for  the  groups  in  three  variables  (§  75).  How 
ever,  we  shall  begin  with  the  primitive  simple  groups,  the 
construction  of  which  is  the  most  difficult  problem  in 
the  present  chapter;  and,  proceeding  at  a  comparatively 
slower  pace  while  dealing  with  these  groups,  we  shall 
determine  completely  the  generating  transformations  of 
every  type  under  this  head. 

On  the  other  hand,  there  are  a  host  of  types  of  the 
somewhat  more  easily  constructed  groups  in  four  variables 
which  are  non-primitive  or  contain  non-primitive  invari 
ant  subgroups.  We  shall  therefore  not  attempt  to  list 
all  of  these  groups,  but  shall  give  an  outline  of  the  theory 
and  so  much  of  the  detail  that  the  student  may  encounter 
no  serious  trouble  in  constructing  such  of  these  groups  as 
may  be  needed  for  any  purpose. 

The  following  propositions  are  of  constant  use  and 
may  be  proved  by  the  student  (cf.  §  31,  Exercise  3): 

1°.  A  set  of  conjugate  operators  of  a  group  G  generate 
an  invariant  subgroup  of  G. 

2°.  If  the  generators  of  G  all  possess  an  invariant 
configuration  (function,  equation,  point,  line,  plane,  etc.), 
then  G  will  possess  the  same  invariant  configuration. 

For  example,  to  examine  if  G  is  monomial,  it  is  sufficient  to  try 
out  the  generators  of  G  for  an  invariant  set  of  four  planes 
XiXzXzX^Q.  (Cf.  §77,  where  the  solution  of  this  problem  is 
indicated  for  a  group  in  three  variables.) 

139 


140  FINITE  COLLINEATION  GROUPS 

Again,  to  examine  if  G  is  intransitive  or  imprimitive  in  two  sets 
of  two  variables  each  (cf.  §  120),  we  write  down  two  sets  of  two 
linear  functions  of  the  variables  with  undetermined  coefficients  : 


The  conditions  that  each  generator  of  G  will  transform  these  sets 
as  sets  of  intransitivity  or  imprimitivity  will  furnish  a  number  of 
equations  among  the  16  coefficients  a\  .  .  that  must  be  simultane 
ously  fulfilled.  Thus,  if  a  generator  A  permutes  the  two  sets,  the 
expressions  in  the  first  set  are  by  A  transformed  into  linear  functions 
of  the  expressions  in  the  second  set,  and  vice  versa;  that  is,  certain 
constants  \,  .  .  ,  ^4  can  be  found  such  that  the  following  identities 
in  xi,  x2,  x3,  x*  are  fulfilled: 


It  is  apparent  that  the  labor  involved  in  a  problem  of  this  kind 
is  somewhat  tedious,  but  is  not  difficult. 

Notation.  —  Throughout  this  chapter  the  letters  i  and  o> 
represent  the  primitive  fourth  and  third  roots  of  unity, 
i  =  l/^l,  a>=(-l+iV3)/2,  respectively. 

Multipliers.  —  We  shall  often  have  occasion  to  refer  to 
the  multipliers  or  characteristic  roots  of  a  transforma 
tion  (§§  2,  23).  These  multipliers  are  in  such  cases 
inclosed  in  brackets:  [a,  ft,  y,  8].  Thus,  the  multipliers 
of  the  transformations  A\,  A2,  A3,  A4  of  (9),  §  123,  are 
all  as  follows:  [1,  1,  -1,  -1].  We  call  to  mind  that 
conjugate  transformations  have  the  same  multipliers 
(§  23)  ;  hence  if  a  group  G  contains  a  Sylow  subgroup  of 
order  7  generated  by  a  transformation  whose  multipliers 
are  [1,  jS,  04,  /32],  the  multipliers  of  any  transformation  in 
G  of  order  7  are  either  [1,  /?,  £4,  ft2]  or  [1,  £3,  ft5,  ft6]  (§  36). 

Type.  —  As  hitherto,  any  one  of  a  series  of  equivalent 
groups  (§51,  4°)  may  be  selected  as  a  type  of  the  groups 
of  the  series.  One  group  may  thus  have  the  type  of 
another  group  without  being  written  in  the  same  form. 
For  instance,  the  group  (A),  §  102,  has  10  subgroups  all 
of  type  (c),  §  111,  though  only  one  of  them  has  the 


LINEAR  GROUPS  IN  FOUR  VARIABLES  141 

canonical  form.  It  also  possesses  6  subgroups  of  type 
(g),  none  of  which  have  the  monomial  form  in  the  variables 
chosen  for  (A). 

There  are  in  all  30  types  of  primitive  groups  in  four  variables. 
These  types  are  here  designated  as  follows:  (A)  —  (F),§102;  (G)  — 
(K),§119;  l°-7°,  §121;  8°-12°,  §122;  and  13°-21°,  §124. 

THE   PRIMITIVE   SIMPLE    GROUPS,    §§  102-117 

102.  List  of  the  groups.  The  groups  (A)-(D)  are 
isomorphic  with  the  alternating  substitution  groups  on 
5-7  letters:  abcdefg.  The  first  three  are  constructed  by 
means  of  Moore's  theorem  (§50;  cf.  §§58,  82),  which 
gives  two  solutions  for  the  group  of  order  60<£.*  The 
groups  (D)-(F)  are  obtained  in  §§  106-109,  115,  116. 

(A)  Group  of  order  60<£  generated  by  F1}  F2,  Fs,  where 

Fi-  (1,1,  «*,«*); 


The  corresponding  substitutions  of  the  alternating 
group  are  (abc),  (ab)(cd),  (ab)(de). 

(B)  Group  of  order  60  generated  by  FI,  Fi,  F'3,  where 


*Seo  Maschko,  Mathematische  Annalen,  LI  (1899),  278-89.  A 
number  of  the  types  given  by  Maschke  are  intransitive.  To  obtain  the 
group  (D)  by  Moore's  theorem  wo  should  have  to  start  with  one  of  these 
intransitive  groups. 


142  FINITE  COLLINEATION  GROUPS 

(C)  Group  of  order  360<£  generated  by  FI,  F2,  F3  of 
(A),  and  F4:  Xi  =  xi,  x2  =  x[,  z3=  -x'4,  z4  =  -x'3. 

The  corresponding  substitutions  of  the  alternating 
group  are  (abc),  (ab)(cd),  (ab)(de),  (ab)(ef). 

(D)*  Group  of  order  J  •  7  ty  =  2520<£  generated  by 
S,  T,  W,  where 

ft-  ft  A  Aft  i 

T:  Xi  =  x(,  xz  =  Xs,  x3  =  Xt,  X4  =  xi; 
W:  Zi  =  m(p24+^+Z3+4),  X2  =  m(x'l-qx2-pxf,-px'4)) 


where 

The  corresponding  substitutions  of  the  alternating 
group  are  (abcdefg),  (bce)(dgf),  (bce)(dfg). 

(E)f  Group  of  order  168<#>  generated  by  S,  T  of  (D), 
and 
R  :  zi  =  n 


where  n  =  77s,  « 
v  i 

(F)J  Group  of  order  26-34-.5<£  =  25920<£  generated  by 
T  of  (D)  and  C,  D,  7,  F,  where 

C  =  .(1,1,0,,^);   D=Ku,,u>,  1); 

F:  x^xl,  5Ct-Jb(a4-ha4-«-aO,  z3  =  /K^+" 


F:  Xi=-4,  aj2  =  4,  2:3=-^,  x4=-^. 

*  Cf.  Klein,  Mathematische  Annalen,  XXVIII  (1887),  519;    Maschko 
•bid.,  LI  (1899),  p.  291. 

f  Cf.   Maschke,    Papers   of  the    International    Mathematical    Congress, 
Chicago,  1896,  p.  176. 

t  Jordan,  Traite  des  Substitutions,  Paris,  1870.  318;  Maschke,   Mathe 
matische  Annalen,  XXXIII  (1889),  320. 


LINEAR  GROUPS  IN  FOUR  VARIABLES  143 

103.  Geometrical  properties  of  transformations  of 
variety  2.  There  are  two  kinds  of  transformations  of 
variety  2,  according  to  the  way  in  which  the  multipliers 
are  repeated:  [a,  a,  /?,  /?]  and  [a,  a,  a,  /?]. 

1°.  Interpreting  x\t  x2,  x3,  x4  as  homogeneous  co-, 
ordinates  of  space,  a  transformation  of  the  first  kind, 
written  in  canonical  form  $=(<*,  a,  /?,  /?),  will  leave 
invariant  every  straight  line  of  the  family  axi+bx2  =  Q, 
cxs-\-dx4  =  Q,  where  a,  b,  c,  d  are  arbitrary  constants. 
Correspondingly,  there  is  a  family  of  invariant  lines 
associated  with  any  other  transformation  T  having  the 
same  multipliers  as  S;  and  since  at  least  one  line  can 
always  be  found  belonging  to  two  such  families,  it  follows 
that  S  and  T  have  an  invariant  line  in  common.  If  the 
variables  are  changed  so  that  this  line  is  x\  =  0,  x2  =  0,  the 
group  H  generated  by  S  and  T  is  reducible  (xi  and  x2 
being  transformed  into  linear  functions  of  themselves 
by  both  S  and  T).  Hence,  if  S  and  T  belong  to  a  finite 
group  G,  the  group  H  is  intransitive  (§  20).  Let  its  sets 
of  intransitivity  be  (x\t  x2)  and  (x3,  z4),  and  its  operators 
will  have  the  form  Ci,  §  14.  Then  if  one  of  the  transfor 
mations  is  written  in  canonical  form  as  far  as  the  variables 
of  any  one  set  are  concerned,  the  corresponding  multipliers 
may  be  either  [a,  a],  [/?,  /?],  or  [a,  /?].  In  the  first  two 
cases  S  and  T  are  commutative  (§  6,  Exercise  2).  Accord 
ingly,  if  they  are  known  not  to  be  commutative,  their 
multipliers  must  be  [a,  /?]  for  both  sets  of  intransitivity. 

2°.  The  transformation  A  =  (a,  a,  a,  /?)  leaves  invariant 
every  plane  through  the  point  x\  =  0,  xz  =  0,  £3  =  0  (namely, 
every  plane  whose  equation  is  pxi+ 5x2+^3  =  0).  If  B 
is  a  transformation  having  the  same  multipliers  as  A, 
and  therefore  possessing  a  similar  invariant  configuration, 
the  group  H  generated  by  A  and  B  must  leave  invariant 
every  plane  through  a  certain  line,  namely,  the  line  which 
joins  the  two  invariant  points  in  question.  Moreover,  if 


144  FINITE  COLLINEATION  GROUPS 

A  and  B  belong  to  a  finite  group  G,  and  if  the  variables 
are  so  changed  that  the  invariant  line  is  x\  =  0,  x2  =  0,  the 
group  H  is  intransitive,  its  sets  of  intransitivity  being 
(zi),  fe),  and  say  (z3,  z4).  As  in  1°,  the  multipliers  of 
A  or  B  corresponding  to  the  intransitive  set  (z3,  z4)  are 
[a,  j8],  unless  A  and  B  are  commutative. 

Now  assume  that  A  and  B  are  of  order  5,  so  that  a  and 
ft  are  5th  roots  of  unity.  The  component  of  H  in  the 
variables  (z3,  £4),  which  we  shall  indicate  by  H',  must 
then  be  reducible  to  the  group  (E),  §  58,  by  the  process 
of  §  12.  This  group  can  be  generated  by  the  operators  of 
order  2$  =  4  that  it  contains ;  the  corresponding  operators 
of  H  must  have  the  multiplier  [1]  in  the  sets  (xj)  and  (z2) 
and  will  consequently  have  the  multipliers  [i,  —i]  in  the 
set  (#3,  #4)  (cf.  §  51,  3°).  These  operators  are  accordingly 
of  determinant  unity  in  H';  and  the  group  generated  by 
them,  namely  (E),  is  therefore  a  subgroup  of  H'.  Hence, 
finally,  H'  contains  a  transformation  of  order  3<£,  whose 
multipliers  are  [  — «,  —  w2],  and  correspondingly  H  contains 
a  transformation  whose  multipliers  are  [1,  1,  —  to,  —  o>2] 
and  is  not  primitive  (§  70). 

Next  assume  A  and  B  two  non-commutative  conjugate 
operators  under  G,  of  any  order.  The  line  Xi  =  x2  =  0 
invariant  under  A  and  B  cannot  be  invariant  under  every 
conjugate  to  A,  unless  G  possesses  an  invariant  intransi 
tive  subgroup  (§  101,  1°).  Hence  assume  a  conjugate  C 
which  does  not  leave  this  line  invariant.  There  is,  how 
ever,  an  invariant  plane  common  to  A,  B}  C,  which  passes 
through  the  line  Zi  =  £2  =  0,  as  is  readily  seen;  and  we 
can  so  choose  the  variables  that  this  plane  is  x\  =  0. 
The  group  K  generated  by  A,  B,  and  C  is  accordingly  in 
transitive,  its  sets  of  intransitivity  being  (xi)  and 
(xz,  x3,  z4).  Moreover,  the  component  of  K  in  the  vari 
ables  (x2,  Xzj  z4)  is  not  intransitive  by  virtue  of  the 
assumed  facts  that  A  and  B  generate  a  transitive  group 


LINEAR  GROUPS  IN  FOUR  VARIABLES  145 

in  (x3,  £4)  and  that  C  does  not  leave  invariant  the  plane 
#2  =  0  (or  it  would  leave  invariant  the  line  Xi  =  x2  =  Q). 

EXERCISES 

1.  Prove  that  if  G  contains  two  non-commutative  transforma 
tions  of  order  5,  both  having  the  multipliers  [a,  a,  £,  /3],  and  if, 
furthermore,    the  intransitive  group  generated    (cf.    1°   above)    is 
abelian  in  one  of  the  sets  of  intransitivity  and  not  in  the  other,  then 
G  contains  a  transformation  whose  multipliers  are  [1,  1,—  w,   —  w2] 
and  is  not  primitive. 

2.  If   two  transformations  D  and  A,   whose  multipliers  are 
respectively  K  w,  w,   1]  and  [alt  ait  03,   a3],  where  af  =  a-j=a|  =  l, 
belong  to  a  finite  group  G,  the  subgroup  generated  by  them    is 
either  abelian  or  is  intransitive  in  (1,  3)  variables.     Moreover,  the 
component  group  in  the  intransitive  set  containing  3  variables  is 
again  intransitive,  since  no  finite  transitive  group  in  3  variables 
contains  two  non-commutative  operators  of  orders  5  and  3  respec 
tively,  the  latter  having  the  multipliers  [w,  w,   1].     Hence  prove 
that  either  the  subgroup  of  G  generated  by  all  the  conjugates  to  A 
is  intransitive  (all  being  commutative  with  D),  or  that  G  contains 
a  transformation  whose  multipliers  are  [1,  1,  —  w,  —  w2]  and  is  not 
primitive. 

104.  Theorem  1.  No  primitive  simple  group  G  can 
contain  a  transformation  of  order  5  and  variety  2. 

Since  G  can  have  no  invariant  subgroup  other  than 
itself,  a  conjugate  set  of  operators  of  order  5  must  gener 
ate  G  (§  101,  1°).  The  theorem  now  follows  from  §  103, 
unless  every  pair  of  non-commutative  conjugate  trans 
formations  of  order  5  and  variety  2,  having  the  multi 
pliers  [a,  a,  fi,  /?],  generate  an  intransitive  group  H  in 
two  sets  (xi,  z2),  Oa,  z4),  primitive  in  both  sets. 

Consider  two  such  non-commutative  transformations, 
S  and  T.  The  two  component  groups  in  two  variables 
generated  by  S  and  T  must  each  be  reducible  to  the 
group  (E),  §  58,  by  the  process  of  §  12;  and  it  is  found 
that  these  groups  are  equivalent,  since  the  generating 
transformations  S  and  T  have  the  same  multipliers  [a,  (3] 


146  FINITE  COLLINEATION  GROUPS 

in  both  sets.  An  appropriate  choice  of  variables  will 
therefore  cause  the  corresponding  matrices  to  be  identical, 
so  that  the  transformations  of  H  all  have  the  form 


a  b  0  0 

c  d  0  0 

0  0  a  b 

0  0  c  d 


(1) 

It  is  now  easily  proved  that  each  line  of  the  family 

where  A.  and  /*  are  arbitrary  constants,  is  invariant  under 
H.  A  similar  set  of  lines  will  evidently  be  invariant  under 
the  group  H'  generated  by  S  and  a  third  transforma 
tion  U  having  the  same  multipliers,  unless  S  and  U  are 
commutative.  Assuming  that  the  variables  of  G  are 
selected  so  that  S  has  the  canonical  form  (a,  ft,  a,  (3)}  we 
find  that  after  a  slight  reduction  the  second  set  of  lines 
can  be  defined  by  the  equations 


where  a,  6,  c,  d  are  certain  constants  depending  on  U,  and 
A/,  p'  are  arbitrary  constants. 

The  families  (2)  and  (3)  are  now  seen  to  have  at  least 
one  line  in  common,  namely,  one  for  which  A  =  A',  n  =  nr, 
(\'a+n'c)n=(\'b+[jifd)\.  It  follows  that  the  group 
generated  by  H  and  H'  is  reducible  and  therefore  intransi 
tive.  Accordingly,  any  two  transformations  T  and  C7, 
both  conjugate  to  S  and  non-commutative  with  it, 
generate  with  S  an  intransitive  group  in  two  sets  of  two 
variables  each.  This  group  can  be  none  other  than  H; 
hence,  all  the  conjugates  to  S  fall  into  two  classes:  those 
which  belong  to  //,  and  those  which  are  commutative 
with  S.  Any  transformation  in  the  latter  class  must  be 
commutative  with  every  transformation  of  H,  since  it 


LINEAR  GROUPS  IN  FOUR  VARIABLES  147 

must  have  the  form  of  a  similarity-transformation  for 
each  of  the  two  sets  of  intransitivity :  (a,  a)  in  one  and 
(/?,  /?)  in  the  other. 

The  group  generated  by  S  and  all  its  conjugates  is 
therefore  either  abelian  or  it  contains  the  intransitive, 
group  H  as  an  invariant  subgroup.  In  any  event,  G 
is  not  a  primitive  simple  group. 

105.  Theorem  2.    //  a  Sylow  subgroup  P  of  a  simple 
group  G  is  abelian  and  contains,  aside  from  the  identity, 
operators  of  order  p  only,  where  p  is  a  prime  number,  then  P 
is  invariant  under  a  larger  subgroup  Q  of  G  of  such  a  nature 
that  not  one  of  the  operators  of  P  except  the  identity  is  in 
variant  under  Q. 

This  theorem  is  a  particular  case  of  a  theorem  given 
by  the  author  in  Transactions  of  the  American  Mathe 
matical  Society,  XI  (1910),  2.  The  proof  is  long  and 
will  not  be  reproduced  here;  an  outline  of  the  principles 
involved  is  given  below  (§  107). 

106.  Theorem  3.     The  group  (F),  §  102,  is  the  only 
primitive  simple  group  which  contains  a  transformation  of 
order  3  whose  multipliers  are  [I,  <*>,  o>,  w].* 

Proof. — In  a  group  G,  let  *Si,  Sz,  .  .  ,  Sh  be  a  set  of 
conjugate  transformations  which  have  the  multipliers 
[1,  w,  tu,  w]  and  which  generate  an  invariant  subgroup  H. 
Since  G  is  to  be  a  simple  group,  we  have  H  =  G,  and  H 
cannot  be  intransitive  by  assumption.  Hence  (§  103,  2°), 
we  can  find  three  transformations  among  Si,  .  .  .  which 
generate  an  intransitive  group  in  sets  of  (1,  3)  variables, 
say  (x\),  (xzj  Xsj  x4),  while  the  component  group  in  the 
variables  (z2,  £3,  £4)  is  transitive  as  far  as  these  variables 
are  concerned.  Moreover,  this  component  cannot  be 

*  Bagnera,  Rendiconti  del  Circulo  matematico  di  Palermo,  XIX  (1905), 
19  fl. 


148  FINITE  COLLINEATION  GROUPS 

imprimitive  since  it  is  generated  by  transformations  hav 
ing  the  multipliers  [o>,  w,  1]  (cf.  §  86,  5°). 

Now,  consulting  the  groups  in  three  variables  (chap,  v), 
we  discover  that  only  one  primitive  group  possesses 
operators  of  order  3<£  and  variety  2,  namely,  the  group  (G), 
§  79,  of  order  216<£.  This  group  may  be  written  as  a 
group  K  in  four  variables  so  that  the  operators  just 
mentioned  (namely,  those  conjugate  to  U,  §79)  become 
transformations  having  the  multipliers  [w2,  w2,  w2,  1]  in  K. 
The  similarity-transformation  (w,  w,  w)  in  (G)  becomes 
the  transformation  (1,  w,  w,  o>)  in  K;  the  latter  group  is 
therefore  of  order  216-3  =  648. 

107.  A  Sylow  subgroup  P  of  order  81  is  generated  by 
the  transformations  in  K  corresponding  to  Si,  T,  U 
(§§  77,  79).  It  contains  an  invariant  abelian  subgroup  PI 
of  order  27,  generated  by  Si=(l,  1,  o>,  <o2),  T^SiT^ 
(1,  o>2,  1,  <o),  and  £7=(o>2,  o>2,  o>2,  1);  and  it  follows  from 
the  general  theorem  referred  to  in  §  105  that  G  must 
contain  a  transformation  W\  which  transforms  PI  into 
itself  and  which  transforms  Xi  into  a  variable  different 
from  x\.  The  successive  steps  in  the  proof  of  the  the 
orem  as  applied  to  the  case  under  consideration  are  as 
follows  : 

1°.  The  operators  of  G  transform  the  plane  Xi  =  0  into 
a  number  of  distinct  planes,  say  x\  =  0,  x{  =  0,  x"  =  0,  .  .  . 
The  geometrical  configuration  composed  of  these  planes: 

.  .    =0 


is  transformed  into  itself  by  G;  i.e.,  J  is  an  invariant  of  G. 
Moreover,  since  G  is  simple,  J  must  be  an  absolute  invari 
ant  (§  88),  since  otherwise  G  is  isomorphic  with  an  abelian 
group  in  one  variable,  namely,  the  variable  J. 

2°.  Now  let  the  planes  (factors)  in  J  be  separated  into 
sets  such  that  planes  of  any  one  set  are  obtained  one  from 
another  by  operating  by  the  transformations  of  PI.  The 


LINEAR  GROUPS  IN  FOUR  VARIABLES  149 

number  of  planes  in  a  set  is  a  factor  of  27,  and  the  sets 
do  not  overlap.  If  TT  is  a  subproduct  of  J,  made  up 
of  the  factors  of  a  set,  then  TT  is  an  invariant  of  PI.  From 
this  we  find  that  if  *  contains  only  one  factor,  it  is  one  of 
the  variables  x\  x2,  x3,  x*.  If  it  contains  3,  9,  or  27 
factors,  then  (TT)U  =  TT. 

3°.  The  plane  #i  =  0  constitutes  a  set  by  itself,  and 
(xi)  U  =  <»2Xi.  But,  since  J  is  an  absolute  invariant 
under  G,  we  must  have  ( J)  U  =  J.  It  follows  that  there  is 
at  least  one  subproduct  •*  different  from  x\  and  such  that 
(TT)U  =  CTT,  where  c?^l.  By  2°  this  factor  TT  is  one  of  the 
variables  xz,  xs,  z4.  Assuming  that  W  was  the  trans 
formation  of  G  which  transformed  x\  into  TT,  we  find 
that  the  group  WPiW~l  leaves  Xi  =  0  invariant  (viz., 
(a?i)TFPiTr-1«(»)PiTr-l-(c'»)TF-1=c/a;i)  and  will  there 
fore  generate  with  K  an  intransitive  group  in  (1,  3)  vari 
ables.  However,  since  (G),  §  79,  cannot  be  a  subgroup  of 
a  larger  group  in  three  variables,  except  one  obtained  by 
adding  new  similarity-transformations,  it  follows  that 
WP\W~l  is  a  subgroup  of  K. 

4°.  By  means  of  Theorem  7,  (6),  §  36,  we  can  now 
find  a  transformation  L  in  K  which  transforms  the  Sylow 
subgroup  to  which  PI  belongs  into  the  Sylow  subgroup 
to  which  WP\W~ 1  belongs.  Since  a  given  Sylow  subgroup 
of  order  81  contains  a  single  abelian  subgroup  of  order  27, 
it  follows  that  L~lPlL  =  WPiW~1.  The  transformation 
LW  is  now  found  to  possess  the  properties  demanded  of  W i 
above. 

108.  The  conditions  W^lPiWi  =  Pi,  (xi)Wi  =  *,  imply 
that  Wi  has  the  monomial  form  and  transforms  x\  into 
one  of  the  variables  z2,  £3,  z4,  multiplied  by  a  constant. 
In  K  we  have  already  the  monomial  transformations  T 
and  F2  which  permute  the  variables  as  do  the  substitu 
tions  (xi)(x2xsx4)  and  (xi) (x2) (x3x4)  respectively  (§41). 
It  is  therefore  possible  to  multiply  W\  by  such  a 


150  FINITE  COLLINEATION  GROUPS 

transformation  in  the  group  generated  by  T  and  V2  that 
this  product  has  the  form 

F'    x\  =  ttx'i}  #2  —  6#2,  Xz  =  cx\,  x\  =  dx[. 

Now,  an  odd  power  of  F  has  the  same  monomial 
form;  we  may  therefore  replace  this  transformation  by 
such  an  odd  power  of  itself  that  the  new  F  is  of  order  2". 
Furthermore,  since  F2  =  (ac,  62,  ac,  d2)  and  is  now  by 
assumption  of  order  2n~l,  and  since  no  transformation  in 
canonical  form  (62,  ac,  d2)  belongs  to  (G)  except  such  as 
are  of  order  3<£,  it  follows  that  ac  =  b2  =  d2.  In  addition, 
the  determinant  of  F  is  —  acbd=  1.  Consequently 

(4)  64==Fl,  d  ==»=&. 

To  determine  the  coefficients  more  fully  we  construct 
the  pr6duct  FV.  Its  characteristic  is  (6+do>)/i/—  3  = 
6(1  —  o>)/3  or  =  &w2,  according  as  d  =  b  or  =  —b.  But 
6(1  —  <i>)/3  cannot  be  written  as  a  sum  of  four  roots  of 
unity  (§86;  §  133,  6°)  for  the  values  of  b  satisfying  (4). 
Hence  we  have  d  =  —  b;  and  we  may  now  multiply  F  by 
such  a  power  of  (z,  z,  z,  i)  that  we  obtain  6=1,  d=  —  1, 
ac=l.  Finally,  the  change  of  variables  indicated  by  the 
transformation  (  —  a,  1,  1,  1)  (cf.  §  13)  does  not  alter 
the  group  K,  while  it  gives  to  F  the  form  as  listed  in 
the  group  (F),  §  102.  Here  we  have  written  C  and  D 
for  Si  and  U2. 

109.  The  group  (F)  is  evidently  primitive  (cf.  §  101, 
2°).  Concerning  the  statement  that  it  is  a  simple  group 
we  merely  observe  that  it  is  not  obtained  by  enlarging 
any  of  the  other  groups  in  this  chapter.  There  remains 
to  prove  that  the  transformations  T,  C,  D,  F,  F  actually 
generate  a  group  of  order  25920<£.  A  group  of  this  order 
cannot  be  a  subgroup  of  a  larger  primitive  simple  group 
(§  111);  hence  no  further  generators  can  be  added  to  the 
list  (F). 


LINEAR  GROUPS  IN  FOUR  VARIABLES  151 

The  40  planes 

xi  =  0,  x2  =  0,  z3  =  0,  x4  =  0; 

(5)          xi-0ix2+02x4=o,  si 

xi-  01*4+02*3  =  0;  (0i,  02  =  1,  «>,  "2), 

are  permuted  among  themselves  by  each  of  the  generating 
transformations  given,  and  therefore  by  (F)  itself.  This 
group  is  accordingly  isomorphic  with  a  (transitive) 
substitution  group  on  40  letters,  and  the  order  of  (F)  will 
be  40A:',.  where  k'  represents  the  order  of  that  subgroup 
K'  of  (F)  which  leaves  Xi  =  0  invariant  (§  45).  We  now 
write  down  a  matrix  with  arbitrary  elements  M=[ast]  to 
represent  a  transformation  in  the  group  K'.  The  con 
ditions  (xi)M  =  mxi,  and  that  M  is  unitary  (§  19)  due  to 
the  fact  that  the  Hermitian  invariant  of  (F)  can  be  none 
other  than  0:1X1+0:2X2+0:3X3+0:4X4,  will  give  us  the  form 
<72,  §  14,  for  M  .  We  finally  impose  the  condition  that 
M  permutes  among  themselves  the  planes  (5)  ;  this  prob 
lem  can  be  simplified  by  multiplying  M  by  suitable  trans 
formations  of  K..  For  instance,  the  case  (x2)M  =  ra'x3  is 
reduced  to  the  case  (x2)M  =  ra'x2  by  substituting  MT~l 
for  M  at  the  outset.  There  results  that  M  belongs  to  K, 
so  that  K'  =  K,  and  the  order  of  (F)  is  25920<£. 

110.  Theorem  4.  No  primitive  simple  group  can  con 
tain  a  subgroup  "Hp"  (§  66). 

Let  G  be  a  group  which  is  shown,  by  application  of  the 
theorems  of  §§  66-68,  to  contain  an  invariant  subgroup 
"Hp".  Then  since  G  is  here  assumed  simple,  we  must 


If  p  =  3,  it  follows  from  §  67  that  a  transformation  T 
whose  order  k  is  prime  to  3  must  have  the  multipliers 
[1,  a,  a,  a];  a*=l.  But  the  determinant  of  this  trans 
formation,  a3,  cannot  be  unity.  Accordingly,  a  group  Hz 
can  contain  no  operator  whose  order  is  prime  to  3,  and  is 


152  FINITE  COLLINEATION  GROUPS 

therefore  not  primitive  (§  61).  Similarly,  a  group  Hp 
is  not  primitive  if  p>3. 

Consider  finally  a  group  H2.  By  §  67,  a  transforma 
tion  in  this  group  of  odd  order  q  must  have  the  multi 
pliers  [a,  a,  ft  fS\.  Therefore,  by  §73,  q<6.  Again, 
by  Theorem  1,  q<5.  The  order  of  H2  is  therefore  limited 
to  the  numbers  2a-36,  and  the  group  is  not  simple  (§  100). 

111.  The  Sylow  subgroups.  The  theorems  of  chap,  iv 
and  the  Theorems  1-4  above  enable  us  to  construct, 
largely  by  trial,  a  Sylow  subgroup  P  of  order  pa<f>  con 
tained  in  a  primitive  simple  group  G.  Thus,  to  con 
struct  a  group  of  order  2a<£  we  make  use  of  the  facts  that 
it  can  have  no  transformation  of  order  8<£  and  variety  4 
or  3  (§  66),  and  none  of  order  4<£  and  variety  2.  More 
generally,  by  following  the  principles  of  §§  66,  68  we  may 
prove  that  a  group  in  which  are  present  the  two  commuta 
tive  transformations  (iy  —i,  1,  1),  (i,  1,  —i,  1),  either 
as  here  written  or  multiplied  by  similarity-transformations, 
must  contain  an  invariant  group  "//2".  By  trial  we 
now  find  that  an  abelian  group  of  order  24<£  will  always 
contain  the  invariant  group  "#2";  hence  (§74),  a^ 
3+3  =  6. 

When  p>3  the  group  P  is  abelian,  and  Theorem  2, 
§  105,  can  be  applied.  For  instance,  let  us  assume  a  group 
P  of  order  7,  generated  by  S=(l,  ft  ft  /?5);  fP=l.  By 
Theorem  2  there  must  be  an  operator  T  which  transforms 
S  into  a  power  of  itself:  T~lST  =  Sk,  where  k^l.  But 
no  matrix  T  of  non-vanishing  determinant  exists  satisfying 
the  equation  ST=TSk.  In  this  manner  the  various 
generators  of  order  7  are  excluded  except  the  following: 

(1,  ft,  P,  W,  08,  P,  P,  /35),  (ft3,  P3,  P,  W,  (I,  1,  A  F). 
But  the  last  two  are  eliminated  by  Theorem  8,  §  70. 
Similarly,  every  abelian  group  of  order  72  or  73  is  elimi 
nated. 


LINEAR  GROUPS  IN  FOUR  VARIABLES  153 

The  group  Q  is  generally  determined  by  Theorem  2  and 
§  68.  Thus,  the  cube  of  the  operator  T  which  transforms 
S=  (1,  ft,  ft4,  ft2)  into  a  power  of  itself  (say  $2)  must  be  a 
similarity-transformation.  For  otherwise  we  should  have 
a  subgroup  "Hp,"  since  Tz  is  commutative  with  S.  The 
non-vanishing  elements  in  the  matrix  of  T  can  now  all  be 
made  unity  by  a  fitting  change  of  variables  and  by  multi 
plying  T  by  a  power  of  (i,  i,  i,  i)  . 

In  listing  the  results  we  use  the  following  abbrevia 
tions  : 

3>,  the  group  of  similarity-transformations  contained 

inG; 

P,  the  Sylow  subgroup  of  order  pa; 
Q,  that  subgroup  of  G  which  contains  P  invariantly, 

when  p>3; 

T,  the  transformation  x\  =  x{,  Xz  —  £3,  #3  =  x\,  x\  =  x^  ; 
R,  the  transformation  x\  =  £3,  x^  —  x'^  Xs  =  x!>,  X4=—x(; 
Ri,  the    transformation    Xi  =  ax{,     x2  =  bx^    £3  =  czJ, 

Xi^dx's,  where  a,  b,  c,  d  are  certain  constants; 
V,  an  operator  which  permutes  among  themselves 
the  variables  £3,  x*,  and  transforms  Xi,  Xz  into 
linear  functions  of  themselves; 

Qi,  the  group  of  order  m<f>  generated  by  all  the  trans 
formations  of  G  which  are  not  of  order  5k  and 
which  are  commutative  with  the  transformation 
A  i  under  (f). 

The  letters  a  and  ft  represent  respectively  primitive 
5th  and  7th  roots  of  unity;  y,  8,  (y/^S),  roots  of  index  11; 
and  c,  £,  (e?^£)>  roots  of  index  13. 


P:  Q: 

Group    Order  Generators  Order  Generators 

(a)  2*4>,    (0^6); 

(b)  3,          W          =K<o,<o2,u>*); 
(C)         3,          F,  =(1,1,  cu,  o>2); 


154  FINITE  COLLINEATION  GROUPS 

P:  Q: 

Group    Order  Generators  Order  Generators 

(d)  9,       Fl9  W  =(»,<*,  1,  1); 

(e)  81,       W,  T(§106); 

(f)  5,       A,        =(1,  1,  «,  a*);       10m<£,     P,  *,  V,  Q,', 

(g)  5,          A,  =(a,a»3);          1Qm^      p?  ^  R2. 

(h)  5,  A2;  20<£,  P,  <*>,  #; 

(j)  25,  A!,  A8=(l,a,  l,a<);  15*,  P,  *,  T7; 

(k)  25,  Ai,  A3;  30*,  P,  *,  T7,  fli; 

(1)  7,  S           =  (1,/3,0*,#0;  21*,  P,  *,  !T; 

(m)  7,  £          =  (ft/3«,/?2,/F);  14<#>,  P,  $,  #2; 

(n)  11,  (7          =(y,y10,8,8io);  22</>,  P,  *,  iP; 

(o)  13,  D!  '      =  (c,  e«   ^  (;i2) .  26<#>,  P,  *,  ,R2; 

(p)  13,  D2        =(1,  c,  e3j£9);  39^  P,  $,  T; 

(q)  13,  D8        -(v^^O;  52cA,  P,  <*>,  R. 

It  is  easy  to  prove  that  no  two  groups  of  the  types  (e), 
(f),  (j),  (k),  (1),  (m),  (n),  (o),  (p),  (q)  can  be  subgroups  of 
a  given  group  G  at  the  same  time.  First,  no  primitive  simple 
group  can  contain  a  transformation  whose  multipliers  are 
[w,  w,  w,  1]  and  at  the  same  time  one  whose  multipliers  are 
those  of  Aij  by  Exercise  2,  §  103.  Thus  the  type  (e) 
excludes  the  types  (f),  (j),  and  (k).  Secondly,  no  primitive 
group  can  contain  a  Sylow  subgroup  of  order  q  and  types 
(l)-(q)  and  at  the  same  time  one  of  order  p  or  p2  (p^q) 
and  types  (e),  (f),  (j)-(q).  For  then  we  should  have  a 
transformation  U  of  order  pq  (§  90,  Cor.),  and  therefore 
a  transformation  of  order  p  (viz.,  Uq)  commutative  with 
one  of  order  q  (viz.,  Up).  But  two  such  transformations 
would  imply  a  subgroup  "Hp,"  p^3  (§  68). 

Hence,  the  order  of  a  primitive  simple  group  in  four 
variables  is  g<j>,  where  g  is  a  factor  of  one  of  the  numbers 
28-34-5,  26-32-52,  26-32-5-7,  26-32-5-l.l,  or  26-32-5-.13: 

112.  Reduction  in  the  number  of  types  of  the  Sylow 
subgroups.  The  types  (f),  (j),  (k),  (m),  (n),  (o),  (p),  (q) 


LINEAR  GROUPS  IN  FOUR  VARIABLES  155 

can  be  eliminated  chiefly  by  aid  of  Theorem  4,  §  93,  from 
which  theorem  it  follows  that  if  the  sum  of  the  products  xx 
exceeds  the  order  #</>  of  the  group  G,  then  G  is  intransitive. 
(For  an  illustration,  the  term  xx  corresponding  to  the 
transformation  Z)2  is  (1+e+eH-*9)  (l+el2+£10+£4)  =3.) 

Consider  the  type  (f).  In  the  group  Q  there  are 
5w<£  —  m<f>  =  4m<£  transformations  whose  orders  are  mul 
tiples  of  5.  Two  such  transformations  belonging  to  two 
different  subgroups  conjugate  to  Q  cannot  be  identical, 
since  the  corresponding  transformations  of  order  5  are 
distinct.  The  sum  M  =  ^\x  for  the  transformations 
of  order  5fc  in  G  will  therefore  be  qh,  where  q  represents 
the  sum  of  such  terms  from  Q,  and  h  the  number  of  sub 
groups  conjugate  to  Q,  namely,  </<£/(  10ra<£),  by  §  30, 
Theorem  2'.  The  group  Q  is  intransitive,  in  (2,  2) 
variables:  (xi,  £2),  (#3,  #4);  or  in  (1,  1,  2)  variables. 
Going  over  the  various  possibilities  in  detail  we  find  that 
the  sum  qis  at  least  10ra</>  (cf.  §  94,  Exercise  1).  The  sum 
M  is  therefore  at  least  (g / (Wm)) •  (Wm<j>)  =#<£.  But  this 
number,  together  with  the  product  xx  corresponding  to 
the  identity,  namely,  16,  exceeds  g<j>.  In  this  way  the 
cases  (n)  and  (o)  can  be  disposed  of. 

113.  Consider  next  the  type  (m).  By  §  111  the 
order  of  G  is  in  this  case  a  factor  of  26«32- 5- ?•<#>.  On  the 
supposition  that  the  order  is  divisible  by  5,  the  group 
would  contain  0<£/(10</>)  or  #<£/(20<£)  subgroups  of  order 
5  (type  (g)  or  (h)),  and  </<£/(  14<£)  subgroups  of  order  7. 
But  these  numbers  should  be  of  the  forms  l+5fci  and 
l+7fc2  respectively  (§36).  By  trial  we  find  this  impos 
sible  except  for  the  order  210<£.  But  this  number  does  not 
correspond  to  a  simple  group  (§  48).  Hence,  the  order 
of  G  is  not  divisible  by  5;  and  therefore  no  transitive 
group  isomorphic  with  G  can  be  constructed  in  5  variables 
(§  93,  Cor.),  a  fact  to  be  used  presently. 


156  FINITE  COLLINEATION  GROUPS 

Now  let  G'  be  a  group  equivalent  to  (7,  differing  from 
the  latter  merely  in  being  written  in  the  variables  yi, 
.  .  ,  2/4  instead  of  «i,  .  .  ,  z4.  The  six  functions 

(6)     Vn 


are  readily  found  to  be  transformed  into  linear  functions 
of  themselves  by  the  intransitive  group  made  up  of  G  and 
G'  as  component  groups.  We  can  therefore  regard  these 
six  functions  as  the  variables  of  a  group  H  simply  iso- 
morphic  with  G  and  G'.  The  transformation  B  becomes 
(1,  1,  ft,  /86,  /33,  /34),  and  R2  will  permute  the  variables 
as  indicated  in  the  cycles  (^12)  (^34)  (^23^14X^24^13).  By 
adding  the  terms  xx  corresponding  to  the  transformations 
of  order  7<£  we  find  that  H  is  intransitive.  Hence,  since 
no  transitive  component  can  contain  5  variables  by  what 
has  been  proved  above,  and  since  no  simple  group  in  2  or  3 
variables  contains  an  operator  of  order  7  which  is  trans 
formed  into  its  inverse  by  one  of  order  2  (viz.,  R2),  it 
follows  that  the  components  of  H  are  three  in  number, 
embracing  the  variables  (#12),  (^34),  (^23,  •  •  ,  ^13).  In 
other  words,  the  two  functions  Vi2  =  Xiy^—x^yi,  ^34  =  ^32/4— 
z42/3,  are  invariants  of  H.  But  it  is  easily  proved  that  a 
transformation  in  four  variables  whose  corresponding 
operator  of  H  transforms  each  of  these  functions  into  a 
constant  multiple  of  itself  must  have  the  form  Ci,  §  14. 
The  group  G  is  therefore  intransitive.  —  In  the  case  (q), 
the  group  H  is  first  proved  to  be  intransitive  in  two 
sets  (viz,  #34),  (^23,  .  ,  ^13),  with  R  permuting  the  vari 
ables  in  the  first  set,  and  with  D3  represented  by  the  iden 
tity  (1,  1)  in  this  set.  But  no  group  in  two  variables 
(^12,  VM),  transitive  or  intransitive,  and  of  order  2n,  can 
be  isomorphic  with  a  simple  group  of  order  52kn  (cf.  §  32). 
In  the  case  (p),  the  group  H  is  likewise  intransitive, 
the  sets  now  embracing  (3,  3)  variables:  (^12,  ^13,  ^14), 


LINEAR  GROUPS  IN  FOUR  VARIABLES  157 

fes,  ^34,  *>4i).     But  no  simple  group  in  three  variables  is  of 
order  13fc  (cf.  chap.  v). 

114.  Finally,  assume  that  the  group   G  contains  a 
group  P  of  type  (j)  or  (k).     The  order  of  G  is  then  a 
factor  of  26-32-52->  (§111). 

Two  different  subgroups  of  order  25  cannot  have  a 
transformation  of  order  5  in  common.  For,  let  A  be  a 
possible  common  transformation;  then,  since  both  groups 
are  abelian,  the  group  generated  by  them  contains  A  as  an 
invariant  transformation  and  is  therefore  intransitive.  If 
we  now  chose  for  A  in  turn  each  of  the  transformations 
that  belong  to  P,  we  find  in  every  case  that  the  sets  of 
intransitivity  involve  (2,  1,  1)  variables.  The  component 
group  in  two  variables  must  be  derivable  from  the  group 
(E),  §  58,  and  G  would  contain  a  transformation  whose 
multipliers  are  [  —  w,  —  w2,  1,  1]  and  is  not  primitive 
(cf.  §  103,  2°). 

The  total  number  of  Sylow  subgroups  of  order  25  is 
therefore  of  the  form  1+25&  (§38,  Exercise  2).  Hence 
(l+25/b)-15<£  or  (l+25fc)-30<£  should  be  a  factor  of  the 
order  of  G  (§  30,  Theorem  2').  But  this  is  found  to  be 
impossible. 

Accordingly,  there  remain  only  the  types  (a),  (b),  (c), 
(d),  (e),  (g),  (h),  and  (1)  as  possible  Sylow  subgroups  of  a 
primitive  simple  group  G  in  four  variables. 

115.  The  simple  group  of  order  7-9- k.     Here  we  have 
a  group  Q  of  type  (1),  generated  by  S,  <$,  and  T.     The 
last   transformation   has    the    multipliers    [1,    1,    <o,    o>2] 
(cf.  §  86,  5°),  and  belongs  to  a  Sylow  subgroup  of  order  9 
and  type  (d).     There  is  therefore  in  G  a  transformation 
W  of  order  3  commutative  with  T,  whose  multipliers  are 
[w,  w2,  1,  1].     Adopting  such  variables  for  G  that  Q  has 
the  form  as  given  under  (1),  we  select  a  matrix  with 
arbitrary    elements    to    represent    W   and    impose    the 


158  FINITE  COLLINEATION  GROUPS 

conditions  that  WT  =TW,  and  that  the  characteristics  of 
W,  WT,  WT2  are  1,  -2,  -2  respectively.  These  condi 
tions  determine  a  matrix  with  only  three  arbitrary  ele 
ments  a,  b,  s: 


~-2-3s 

a 

a 

a 

b 

s+l 

s 

s 

b 

s 

s+l 

s 

b 

s 

s 

s+l_ 

The  characteristics  of  SW  and  S*W  are  [SW]=  -2-3s-f 
(s+l)p,  [&W]=-2-38+(S+l)q,  where  p  =  P+P+(P, 
g  =  £3_j_^5_j_^6.  and  they  are  each  the  sum  of  the  four 
roots  of  unity  which  are  the  multipliers  of  the  respective 
transformations.  If  one  of  these  multipliers  is  a  7fch  root 
of  unity  or  is  (more  generally)  of  index  7n,  then  the  four 
multipliers  are  the  same  as  those  of  S  or  a  power  of  S, 
possibly  multiplied  by  a  similarity-transformation,  since 
we  would  otherwise  have  a  subgroup  "Hp"  (cf.  argument 
at  the  end  of  §  111);  that  is,  the  corresponding  character 
istic  is  r(l+p)  or  r(l+q),  where  r==*=l  or  =*=i.  Kron- 
ecker's  theorem  (§  133,  6°)  can  now  be  applied  directly 
to  the  equation  obtained  by  eliminating  s,  and  we  find  the 
following  alternatives : 

I:      [SW]=-(l+q),[&W]=-l; 
II:     [SW]  =  -l,[&W]=-(l+p). 

Selecting  the  first,  we  derive  s=  —  (4+p)/7.  Again, 
from  W3  =  E  we  now  get  a&=  — 1/7;  and  the  change  of 
variables  expressed  by  the  transformation  (V  a/6,  1,  1,  1) 
will  finally  give  us  the  form  of  W  as  written  in  (D),  §  102. 
The  second  alternative  (II)  above  would  furnish  W2 
instead  of  W. 

There  remains  for  us  to  answer  the  following  questions : 
Do  the  transformations  S,  T,  W  generate  a  group  of 


LINEAR  GROUPS  IN  FOUR  VARIABLES  159 

order  7!<£/2,  isomorphic  with  the  alternating  group  on 
7  letters  ?  Can  no  new  generators  be  added  to  S,  T,  W? 
Concerning  the  first  question  we  remark  that  if  a  group 
G'  of  order  7!<£/2,  isomorphic  with  the  alternating  group 
on  7  letters,  can  be  constructed  as  a  linear  group  in  four 
variables  by  Moore's  theorem  (§  50),  then  the  group 
generated  by  S,  T,  W  must  be  equivalent  to  Gf.  For 
three  such  operators  or  their  equivalents  are  evidently 
present  in  G' ;  moreover,  the  alternating  group  on  7  letters 
contains  no  subgroup  of  order  63fc  except  itself.  That 
the  group  G'  exists  has  been  shown  by  Maschke  (cf. 
footnote  to  §  102). 

To  answer  the  second  question  we  observe  that  an 
assumed  primitive  simple  group  generated  by  S,  T,  W 
and  additional  generators  must  contain  (D)  as  a  subgroup 
by  what  precedes,  and  hence  be  of  order  23+c«9'5'7<£ 
(§111);  c^3.  Now  counting  the  Sylow  subgroups  of 
order  5  and  type  (h),  such  groups  being  already  present  in 
(D),  and  of  order  7,  we  may  readily  prove  that  c  =  0  is 
the  only  solution  (cf.  §  113). 

116.  The  simple  group  of  order  7-3 -k.  As  in  §  115, 
we  chose  such  variables  for  the  group  now  under  dis 
cussion  (G)  that  the  subgroup  Q  of  order  21  will  appear  in 
the  form  (1),  §  111. 

In  G  there  is  a  Sylow  subgroup  of  order  3  generated 
by  T,  and  the  order  of  G  is  a  factor  of  26-3-5-7-<£.  The 
number  of  subgroups  of  order  7  must  be  either  8  or  64,  and 
correspondingly  the  order  of  G  is  either  168<£  or  1344$. 
Only  the  former  number  is  the  order  of  a  simple  group 
(§48). 

This  simple  group  can  be  written  as  a  substitution 
group  K  on  8  letters  abcdefgh  representing  its  8  subgroups 
of  order  21  (§  46,  Cor.).  If  a  represents  the  group  Q,  and 
if  S  transforms  the  group  b  into  c,  etc.,  the  operators  S 


160  FINITE  COLLINEATION  GROUPS 

and  T  are  in  K  represented  by  the  substitutions  (bcdefgh) 
and  (cdf)(ehg)  respectively.  An  additional  generator  of 
G  is  obtained  by  constructing  an  operator  R  of  order  2</>, 
which  transforms  T  into  T2  (§  105)  and  whose  substitu 
tion  in  K  is  (a&)  (ch)  (de)  (fg) .  The  conditions  R~1TR  =  T2, 
R2  =  a  similarity-transformation,  give 


R  = 


r  v  v  v 

w  s  t  u 

w  t  u  . 

W  U  S  t 


We  may  assume  v  =  r;  this  is  equivalent  to  changing 
the  variables  in  G  by  means  of  the  transformation 
(v/r,  1,  1,  1). 

To  further  specialize  the  elements  in  R,  we  note  that 
with  each  subgroup  of  order  21  belongs  an  invariant  plane; 
in  the  case  of  a  this  is  £i  =  0,  the  remaining  planes  being 
(xi)R  =  Q,  (xi)RS  =  Q,  .  .  ,  (x1)RSQ  =  0.  The  planes  be 
longing  to  c  and  h  are  (xi)RS=r(xi-\-^x2+^x3-\-l32X4)=0 
and  (xi)RSG=r(xi-\-(36X2+P*X3-\-(3bX4)=Q',  hence,  since 
c  is  transformed  into  h  by  R,  we  have  correspondingly 


where  k  is  an  undetermined  constant.  This  condi 
tion  will  fix  definitely  the  ratios  of  the  elements  in  R; 
adding  the  condition  that  the  determinant  of  R  is 
unity  we  finally  obtain  this  generator  as  listed  under 
(E),  §  102. 

Having  proved  that  no  operator  except  a  similarity- 
transformation  can  leave  invariant  each  of  the  8  invari 
ant  planes,  we  may  show  that  S,  T,  R  generate  a  group  of 
order  168</>,  isomorphic  with  the  simple  group  of  order 
168  (cf.  §  109). 

*  If  c  =0,  the  group  generated  by  S,  T,  R  is  intransitive. 


LINEAR  GROUPS  IN  FOUR  VARIABLES  161 

117.  The  simple  groups  of  order  5k.     We  come  now 
to  the  problem  of  determining  the  primitive  simple  groups 
whose  orders  are  factors  of  26»32-5-<£,  and  in  which  there 
are  subgroups  Q  of  order  10<£  or  20<£  as  listed  under  (g) 
or  (h),  §  111.     Counting  the  Sylow  subgroups  of  order  5, 
we  discover  that  the  orders  of  the  groups  sought  are  all 
<1000.     Hence,  these  groups  are  isomorphic  with  the 
alternating  groups  on  5  and  6  letters  (§  48),  and  can  be 
constructed  by  Moore's  theorem   (§  50).     There  result 
the  types  (A),  (B),  (C)  (§  102). 

118.  Groups  which  contain  primitive  simple  groups  as 
invariant  subgroups.     In  the  construction  of  such  groups 
we  are  aided  materially  by  the  following  theorems. 

THEOREM  5.  //  a  given  primitive  group  G  does  not 
contain  a  subgroup  "Hp"  (§§  66-68),  neither  does  a  larger 
group  K  in  which  G  is  contained  as  an  invariant  subgroup. 

1°.  We  shall  prove  that  %  if  K  contains  an  invariant 
subgroup  "Hp,"  then  G  must  contain  such  a  group  also. 
Let  H  be  a  subgroup  "Hp"  of  K,  and  let  us  assume  to 
begin  with  that  T  is  an  operator  common  to  both  G  and  H 
and  is  not  a  similarity-transformation.  If  then  V  belongs 
to  G,  the  transformation  VT  belongs  to  G,  and  the  equa 
tion  (9),  §  66,  is  true,  since  it  is  true  when  V  is  any  trans 
formation  of  K  and  T  of  H.  But  if  (9)  is  true,  the  group 
G  contains  an  invariant  subgroup  "Hp"  (by  the  argu 
ments  of  §  66),  contrary  to  assumption.  Hence,  G  and  H 
can  have  no  operators  in  common  except  similarity- 
transformations. 

2°.  Now  let  A  be  any  operator  of  G,  and  R  of  the 
group  H,  in  K.  Then  since  H  is  an  invariant  subgroup, 
A~1RA  =  Ri  belongs  to  H.  Again,  since  G  is  an  invariant 
subgroup,  RAR~1  =  Ai  belongs  to  'G.  From  these  two 
equations  we  obtain  RA=AR\  =  A\R,  and  therefore 


162  FINITE  COLLINEATION  GROUPS 

RiR-l  =  A~lAi.  But  R&-1  belongs  to  Hand  A~lAl  to  G. 
It  follows  by  1°  that  RiR~l  =  Ei,  a  similarity-transforma 
tion.  Hence,  R^E^R,  so  that  A~1RA=E1R. 

3°.  Assuming  R  not  to  be  a  similarity-transformation, 
let  it  be  written  in  canonical  form,  and  let  A  represent 
in  turn  every  operator  of  G.  We  can  then  prove  by  the 
process  of  §  61  that  G  is  not  primitive.  Accordingly, 
since  this  violates  the  hypothesis  regarding  G}  the  assump 
tion  that  H  is  contained  in  K  is  untenable. 

THEOREM  6.  Let  G  be  a  self-conjugate  subgroup  of  K 
of  index  h  (§  28)  and  P  a  Sylow  subgroup  of  G.  Further 
more,  let  Q  and  Q'  be  the  largest  subgroups  of  G  and  K 
respectively  which  contain  P  as  an  invariant  subgroup. 
Then  Q  is  a  subgroup  of  Q'  of  index  h. 

Proof. — Let  g<i>  and  gh<l>  be  the  orders  of  G  and  K; 
q  and  q'  the  orders  of  Q  and  Q',  and  n  the  number  of 
Sylow  subgroups  of  the  same  order  as  P  in  G.  These 
subgroups  form  a  single  conjugate  set  (§  36),  and  therefore 
q=(g^)/n.  If  A  is  any  operator  of  K,  then  A~1PA 
belongs  to  G  and  is  of  the  same  order  as  P  (§  30, 
Exercise  2);  the  group  A~1PA  is  therefore  the  group 
P  or  another  Sylow  subgroup  conjugate  to  P.  Hence, 
the  n  subgroups  conjugate  to  P  in  G  also  form  a  single 
conjugate  set  in  K,  and  q' =  (gh<t>)/n,  so  that  qf/q  =  h. 
Finally,  Q  is  evidently  a  subgroup  of  Q'  by  the  definition 
of  these  groups. 

119.  Now,  in  the  case  of  the  Sylow  subgroup  (1),  the 
group  Q  already  has  the  maximum  order  21</>,  barring 
the  existence  of  an  invariant  subgroup  "Hp".  The 
groups  (D)  and  (E),  §  102,  can  therefore  not  be  con 
tained  as  invariant  subgroups  in  larger  groups.  The 
same  argument  applies  to  the  group  (F),  when  for  P 
we  take  a  group  of  order  81,  which  is  already  contained 
in  a  group  Q  of  order  162  generated  by  P  and  V2. 


LINEAR  GROUPS  IN  FOUR  VARIABLES  163 

There  remain  the  groups  (A),  (B),  and  (C).  Taking 
for  P  a  subgroup  of  order  5,  the  group  Q  is  here  of  order 
10<£  (type  (g),  §  111),  and  may  be  enlarged  to  a  group 
Q'  of  order  20<£  (type  (h)).  Hence,  these  groups  (A)-(C) 
may  possibly  be  enlarged  to  groups  of  twice  their  orders. 
In  fact,  the  student  may  verify  that  the  transformation* 

F':     zi  =  «K  z2  =  «K  a*  =  ^J,  *4  =  iK;  *-^, 

v  2 

does  not  belong  to  the  group  (B),  but  will  transform  the 
generators  of  (B)  in  the  same  manner  as  the  substitution 
(ab)  will  transform  the  corresponding  substitutions. 
Again,  the  transformation* 

F": 


does  not  belong  to  either  (A)  or  (C),  but  will  transform 
the  generators  there  listed  into  new  generators  in  the  same 
way  as  the  substitution  (ab)  will  transform  the  corre 
sponding  substitutions.  We  therefore  obtain  three  new 
groups,  respectively  isomorphic  with  the  symmetric 
groups  on  5,  5,  6  letters,  namely, 

(G)  Group  of  order  120<£  generated  by  (A)  and  F". 

(H)  Group  of  order  120<£  generated  by  (B)  and  F'. 

(K)  Group  of  order  720</>  generated  by  (C)  and  F". 

Evidently  none  of  these  new  groups  can  be  enlarged. 

NON-PRIMITIVE    GROUPS    AND     PRIMITIVE     GROUPS    WHICH 

CONTAIN  INVARIANT  NON-PRIMITIVE  SUBGROUPS, 

§§  120-125 

120.  Intransitive  and  imprimitive  groups.  As 
remarked  in  §  101,  we  shall  not  present  an  exhaustive 
analysis  of  the  remaining  groups  in  four  variables.  The 
problems  involved  are  not  very  difficult,  but  need  on 
occasion  a  mass  of  painstaking  labor. 

*  See  Maschko,  Mathematische  Annalen,  1899,  referred  to  in  footnote 
to  §  102. 


164  FINITE  COLLINEATION  GROUPS 

The  intransitive  groups  in  four  variables  fall  into  four 
classes,  according  to  the  number  of  variables  in  the 
different  sets  of  intransitivity,  namely,  (1,  1,  1,  1),  (1,  1,  2), 
(1,  3),  or  (2,  2)  variables.  To  construct  such  a  group, 
say  one  whose  sets  of  intransitivity  involve  (2,  2)  variables : 
(xi,  xz)  and  (#3,  #4),  we  select  two  transitive  binary 
groups  (chap,  iii,  (B)-(E))  that  can  be  written  as  iso- 
morphic  groups  directly  or  after  being  enlarged  by  the 
method  of  §  85.  To  the  identity  of  one  group,  repeated 
say  k  times,  will  correspond  an  invariant  subgroup  of  the 
other  of  order  k  (cf.  §  32).  The  transformations  of  these 
groups  separately  need  not  be  of  determinant  unity  (cf . 
§  51,  2°).  Hence,  the  matrices  of  the  groups  (B)-(E) 
may  first  be  modified  by  multiplying  them  by  certain 
similarity-transformations  (cf.  §§  10,  12);  moreover, 
operators  in  the  canonical  form  (a,  a,  ft,  /3),  where  a2/22  =  1, 
may  even  be  added  as  new  generators. 

Imprimitive  groups  are  of  two  kinds:  (a)  groups 
which  have  the  monomial  form  by  a  proper  choice  of 
variables,  and  (6)  groups  of  the  form  discussed  at  the 
opening  of  §  60.  A  group  G  of  the  first  kind  (a)  is  iso- 
morphic  with  one  of  the  transitive  substitution  groups 
on  four  letters,  namely,  the  alternating  or  symmetric 
groups  on  four  letters,  the  Sylow  subgroup  of  order  8  of 
the  symmetric  group,  or  the  group  (8),  §  43.  Such  a 
group  is  therefore  of  order  120,  240,  80,  or  40,  where  0 
represents  the  order  of  the  invariant  abelian  subgroup 
which  has  the  canonical  form  when  G  is  written  in  mo 
nomial  form.  A  group  of  the  second  kind  (6)  is  generated 
by  an  invariant  intransitive  subgroup  H  and  a  trans 
formation  T  which  permutes  the  two  sets  of  intransi 
tivity  of  H.  If  the  variables  are  so  chosen  that  the 
matrices  of  H  have  the  form  C\,  §  14,  that  of  T  will  have 
the  form  of  the  second  matrix  given  in  §  60. 


LINEAR  GROUPS  IN  FOUR  VARIABLES  165 

To  evaluate  the  elements  of  T  more  definitely  we 
may  proceed  as  follows.  The  Sylow  subgroups  of  H  of 
order  3a  must  be  permuted  among  themselves  by  T,  and 
since  they  already  constitute  a  single  conjugate  set  under 
H,  it  is  possible  to  find  an  operator  L  in  H  such  that  the 
product  LT  transforms  a  given  Sylow  subgroup  P  into 
itself  (cf.  §  107,  4°).  Writing  P  in  canonical  form,  we  find 
that  the  new  T  (viz.,  LT)  is  a  monomial  transformation, 
and  we  can  even  make  such  additional  changes  in  the 
variables  that  T  has  the  form  Xi  =  x'^  x2  =  x\,  and  either 
x3  =  ax{,  X4  =  (3x!>,  or  x$  =  a.x'>,  X4  =  /3x[.  Moreover,  since 
any  odd  power  of  T  permutes  the  two  sets  of  intransitivity 
of  H  also,  we  may  assume  that  originally  such  a  power 
had  been  selected  that  the  new  T  is  of  order  2b. 

EXERCISE 

Prove  that  either  (I),  T  has  the  form  Xi  =  x'3,  x2=x'i,  xz  =  a-x{, 
xt  =  ax2,  or  else  (II),  to  the  group  of  similarity-transformations  of 
one  of  the  components  of  //  embracing  one  set  of  intransitivity 
(xi,  £2)  will  correspond  an  invariant  subgroup  of  the  other  set 
(x3,  xt)  of  order  120,  240,  or  600.  (Hint:  construct  T2  and  T~1PT, 
belonging  to  H,  and  make  use  of  the  facts  that  none  of  the  groups 
(C)-(E),  chap,  iii,  have  a  transformation  of  order  20  commutative 
with  one  of  order  30;  and  also  that  if  H  contains  a  transformation 
whose  multipliers  are  [0,  9,  pw,  pw2J  and  is  of  order  0  in  the  variables 
(xi,  z2)  and  of  order  30  in  the  variables  (x3,  xt),  then  (II)  is  true.) 

121.  Primitive  groups  having  invariant  intransitive 
subgroups.*  It  is  easily  proved  that  if  a  primitive  group 
G  contains  an  invariant  intransitive  subgroup  H,  none 
of  the  sets  of  intransitivity  can  embrace  just  one 
variable  (cf.  method  of  proving  the  lemma,  §  61).  The 
transformations  of  H  must  therefore  have  the  form  of 

*  Goursat  has  determined  all  the  groups  in  four  variables  which  leave 
invariant  the  quadric  xixt  —  x-2X3=0.  These  include  all  the  groups  enumer 
ated  in  this  and  the  following  paragraph.  Soo  "Sur  les  substitutions 
orthogonales,  etc.,"  Annales  scientifiques  de  V&cole  Normale  Superieure, 
(3),  VI  (1889),  pp.  02-79. 


166  FINITE  COLLINEATION  GROUPS 

Ci,  §  14,  and  the  two  straight  lines  Xi  =  Xz  =  Q,  x3  =  x4  =  0 
are  invariant  under  H.  If  these  are  the  only  invariant 
lines,  G  will  be  intransitive  or  imprimitive,  since  a  line 
which  is  invariant  under  H  is  by  an  operator  of  G  trans 
formed  into  a  line  also  invariant  under  H.  We  therefore 
assume  at  least  one  additional  invariant  line;  this  can  be 
written  Xi+Xz  =  Q,  x2+x4  =  0  by  a  suitable  choice  of 
variables.  The  matrices  of  H  are  then  seen  to  have  the 
form  of  the  matrix  (1),  §  104. 

All  the  lines  invariant  under  H  now  belong  to  the 
family  (2),  §  104,  and  are,  as  remarked,  permuted  by  G. 
In  order  that  this  condition  may  be  fulfilled,  the  matrices 
of  G  must  necessarily  have  the  form 


(7) 


pt  pu  qt  qu 

pv  pw  qv  qw 

rt  ru  st  su 

rv  rw  sv  sw 


The  numbers  p,q,..,wm  the  different  transformations 
of  G  are  readily  determined  from  the  fact  that 


are  corresponding  matrices  of  two  groups,  say  G\  and  Gz, 
in  two  variables,  simply  isomorphic  with  G.  (This  we 
discover  when  we  multiply  together  two  transformations 
of  the  form  (7)).  Moreover,  G\  and  G2  may  simultane 
ously  be  given  any  convenient  forms,  independently  of 
each  other,  that  would  be  obtainable  through  separate 
changes  of  variables.  They  must  both  be  primitive, 
and  are  therefore  to  be  selected  from  the  groups  (C)-(E), 
chap.  iii.  For  if,  say,  Gz  were  imprimitive,  we  could 
write  it  in  monomial  form;  the  group  G  would  then 
appear  as  an  intransitive  or  imprimitive  group. 


LINEAR  GROUPS  IN  FOUR  VARIABLES  167 

To  the  group  of  similarity-transformations  in  GI  will 
correspond  an  invariant  subgroup  H2  of  (72,  and  vice 
versa.  The  group  H2  is  transitive,  since  it  includes  a 
component  of  H.  Hence,  if  G%  is  of  order  120,  Hz  is  of 
order  40  (generated  by  W\  and  Wz,  §  57)  or  of  order  120; 
if  the  order  of  G2  is  240,  that  of  #2  is  40,  120,  or 
240;  finally,  if  the  order  of  G2  is  600,  that  of  #2  is  600. 
Setting  up  an  isomorphism  with  the  primitive  group  G\, 
we  find  the  orders  of  the  respective  groups  as  follows  : 

joo  2°°  1°  2°  3°  4°  5°  6°  7° 
Hi:  40  40  120  120  120  120  240  240  600 
Gi:  120  24<#>  12<#>  24<£  12</>  12<A  24</>  24<#>  60</> 
H2:  4</>  4<#>  12<#>  12*  24c#>  600  240  600  600 
G2:  120  240  120  240  240  600  240  600  600 
G:  480  960  1440  2880  2880  7200  5760  14400  36000 

The  groups  1°°  and  2°°  are  monomial,  while  the  groups 
l°-7a  are  all  primitive.  No  new  types  are  obtained  by 
interchanging  the  subscripts  1  and  2  in  3°,  4°,  or  6°,  since 
the  groups  so  derived  are  equivalent  to  the  original 
groups. 

The  primitive  groups  l°-7°  leave  invariant  the  quadric  surface 
£1X4—  x2x3=Q,  and  this  one  only.  The  family  (2),  §  104,  constitutes 
one  of  the  two  systems  of  straight  lines  lying  upon  this  surface;  the 
other  is 


122.  Groups  containing  as  invariant  subgroups  the 
primitive  groups  of  §  121.  An  operator  which  leaves 
invariant  each  of  the  two  families  of  lines  lying  on  the 
quadric  ZiZ4  —  z2z3  =  0  will  have  the  form  (7);  one  which 
interchanges  them  will  have  the  form 

pt  qt  pu  qu 

pv  qv  pw  qw 

rt  st  ru  su 

rv  sv  rw  sw 


168  FINITE  COLLINEATION  GROUPS 

New  groups,  different  from  those  already  listed  in 
§  121,  can  therefore  be  obtained  only  by  including  a 
transformation  T  of  the  form  (8)  among  the  generators 
of  the  groups  l°-7°.  This  operator  will  transform  H\ 
into  Hz  and  vice  versa;  the  orders  of  Hi  and  H2  are  there 
fore  equal,  and  we  are  limited  to  the  cases  1°,  2°,  5°,  and  7°. 

The  order  of  T  may  be  assumed  a  power  of  2  (cf  .  argu 
ment  at  the  end  of  §  120).  Under  this  assumption  it  is 
contained  in  a  Sylow  subgroup  P  of  order  2a+1  (§  39), 
namely,  the  group  generated  by  T  and  the  Sylow  subgroup 
of  G  of  order  2a  which  is  transformed  into  itself  by  T. 
Writing  P  in  monomial  form,  we  find  that  for  T  may  be 
chosen  a  generator  which  permutes  the  variables  in  the 
order  (xi)(x2x3)(x^),  say 

T:    xi  =  ax(,    z2  =  #4    z3  =  y4,    x±  =  %x\      (aS  =  /3y). 

Now  consider  a  group  generated  by  T  and  the  group 
2°.  The  groups  HI  and  H2  being  given  as  in  §  57,  (C),  the 
conditions  that  T2  belongs  to  G  and  that  T~lHiT  =  H2, 
are  now  found  to  be  equivalent  to  the  equations  a4  =  /?4  = 
y4  =  84.  But  since  the  change  of  variables  indicated  by 
the  transformation  (1,  1,  i,  i)  produces  the  operators  of  G 
over  again  in  the  same  forms  (though  not  in  the  same 
order),  we  may  introduce  this  change  (if  necessary)  in  T 
so  as  to  obtain  a2  =  /J2.  Again,  if  a  =  —  ft,  we  may  replace 
T  by  the  product  WT,  where  W  belongs  to  P  and  has  the 
canonical  form  (1,  —1,  1,  —1);  if  a=—  y,  we  replace  T 
by  TW.  Hence,  finally,  we  have  the  two  possibilities: 


There  result  two  groups,  8°  and  9°,  both  of  order  576</>, 
generated  respectively  by  2°  and  T\t  2°  and  T2. 

The  transformation  T$  is  not  contained  in  either  1°  or 
7°.  Hence  we  here  obtain  only  one  new  group  in  each 


LINEAR  GROUPS  IN  FOUR  VARIABLES  169 

case,  10°  and  11°,  of  orders  288<£  and  7200<£,  generated 
respectively  by  1°  and  Ti,  7°  and  TV 

Finally,  the  group  5°  already  contains  the  transforma 
tion  R=  (1,  1,  i,  i)  =  TzT-i1.  The  groups  generated  by  5° 
and  TI  or  by  5°  and  T2  are  accordingly  identical,  furnishing 
a  single  new  group,  12°,  of  order  1152<£. 

EXERCISES 

1.  The  primitive  groups  1°-12°  all  leave  invariant  the  surface 
XiX*— £2X3  =  0,  and  only  this  one  of  the  second  degree.     Prove  that 
any  operator  which  transforms  this  surface  into  itself  must  have  the 
form  (7)  or  the  form  (8).     Hence  prove  that  a  group  which  contains 
self-con jugately  any  one  of  the  groups  1°-12°  is  already  included 
in  this  list. 

2.  The  groups  1°-12°  and  the  group  (B),  §  102,  are  the  only 
primitive  groups  in  four  variables  which  leave  invariant  a  quadric 
surface.     In  the  case  of  the  group  (B),  this  surface  has  for  equation 
Xi+x  1+2x3X4  =  0,    which   can   be   transformed  into   the  equation 
ZiZi  —  Ztz3  =  0    by    the    following    change    of    variables:  Zi  =  z2+z3, 
x2=i(  —  22+33),   x3=  —  Zi,   Xt  =  2zi.     Introduce   this   change  in    (B) 
and  compare  the  new  generators  with  the  matrix  (7). 

123.  Primitive  groups  having  invariant  imprimitive 
subgroups.  In  the  study  of  these  groups  the  following 
proposition  is  found  useful: 

Given  a  group  G  and  a  positive  integer  n.  The  group 
generated  by  the  nth  powers  of  the  operators  of  G  or  of  an 
invariant  subgroup  of  G  is  again  an  invariant  subgroup  of 
G.  (The  method  of  proof  is  embodied  in  Exercise  3,  §  31.) 
It  should  be  noted  that  the  group  generated  by  the  nth 
powers  of  the  operators  of  G  is  not  necessarily  the  group 
generated  by  the  nth  powers  of  the  generators  of  G. 

Now  consider  a  group  G  which  contains  invariantly  an 
imprimitive  subgroup  K  of  the  kind  classified  under 
(6),  §  120.  The  group  generated  by  the  second  powers  of 
the  operators  of  K  is  intransitive,  and  therefore  G  is  found 
among  the  groups  defined  in  §  121.  The  only  exception 


170 


FINITE  COLLINEATION  GROUPS 


is  furnished  by  a  group  K  the  second  powers  of  whose 
operators  all  reduce  to  similarity-transformations..  In 
such  a  case  K  must  be  of  order  2m<£  and  can  therefore  be 
written  in  monomial  form.  We  find  by  trial  that  it  is  of 
order  16<£  and  is  generated  by  the  transformations 


(9) 


.(1,1,  -1,  -1), 

0100 
1000 
0001 
0010 


A2=(l,  -1,  -1,1) 

0010 
0001 
1000 
0100 


to  which  may  be  added  the  similarity-transformation 
(i,  i,  i,  i). 

Next  let  K  belong  to  the  class  (a),  §  120,  at  the  outset. 
The  cases  where  K  is  of  order  4g  or  Sg  come  under  the  class 
of  groups  mentioned  above ;  there  remain  the  cases  where 
K  permutes  the  variables  in  the  same  way  as  the  alternat 
ing  or  the  symmetric  groups  on  four  letters.  The  latter 
case  is  reduced  to  the  former  by  taking  for  a  new  K  the 
group  generated  by  the  second  powers  of  the  transforma 
tions  of  K  as  given.  Again,  when  K  corresponds  to  the 
alternating  group,  it  is  reduced  to  a  monomial  group  of 
order  40  by  taking  the  group  generated  by  the  third  powers 
of  its  transformations. 

Hence,  finally,  our  problem  is  limited  to  that  of  finding 
the  primitive  groups  which  contain  the  group  (9)  self- 
conjugately. 

124.  The  group  F  of  order  11520<t>  isomorphic  with 
the  symmetric  group  on  6  letters.*  Referring  to  §  113, 
let  G  and  G'  be  equivalent  groups  in  x\,  .  .  and  yi,  .  .  , 
and  H  the  isomorphic  group  in  the  six  variables  (6): 

*  Klein,  Mathematischc  Annalen,  II  (1870),  198  ff.;  IV  (1871),  356; 
Maschke,  ibid.,  XXX  (1887),  496  ff. 


LINEAR  GROUPS  IN  FOUR  VARIABLES  171 

Viz,  •  •  ,  ^34.  If  we  in  the  latter  group  change  the  vari 
ables  to  Wi,  .  .  ,  w6,  where 

(10)       JJ1°"11+^"> 

the  group  (9)  will  have  taken  the  canonical  form  and  be  of 
variety  6.  Accordingly,  H  will  permute  among  them 
selves  the  six  variables  wi,  .  .  ,  w6  (cf.  §  61). 

The  following  statements  may  now  be  proved  by  the 
student : 

(a)  A  transformation  of  G  whose  corresponding  trans 
formation  of  H  has  the  canonical  form  must  belong  to  K. 

(b)  Consequently,  if  two  transformations  of  H  per 
mute  Wi,  .  .  ,  w6  in  the  same  way,  one  of  the  two  corre 
sponding  transformations  of  G  is   equal  to  the   other 
multiplied  by  a  transformation  of  K. 

(c)  Now,  the  transformations: 

-i  0  0  i 
0110 
1001 
0  -•  i  0 


/2 


permute  the  variables  Wi,  .  .  ,  WQ  in  the  orders 
and  (wzWtWsWzWs)  respectively,  and  will  therefore  with  K 
generate  a  group  F  of  order  16-720-<£,  isomorphic  with 
the  symmetric  group  on  the  six  letters  w^  .  .  ,  w&. 
By  (a)  and  (6),  this  group  will  contain  every  transforma 
tion  which  leaves  K  invariant. 

(d)  Hence,   finally,   all  the   primitive  groups   which 
contain  K  as  an  invariant  subgroup  are  contained  as 
subgroups  in  the  group '  F. 

(e)  Now,  any  two  subgroups,  F\  and  Fz,  are  equiva 
lent  when  their  corresponding  substitution  groups,   F{ 
and  F-2,  on  the  letters  w\,  .  .  ,  WG  are  conjugate  under 
the  symmetric  group  of  order  720.     For,  if  a  substitution 


172  FINITE  COLLINEATION  GROUPS 

transforms  F{  into  F->,  the  corresponding  operator  of  F 
will  transform  FI  into  F2.  Hence,  to  determine  all  the 
primitive,  non-equivalent  subgroups  of  F,  we  first  select 
a  representative  of  each  set  of  conjugate  subgroups  of 
the  symmetric  group  under  discussion.  Every  such  repre 
sentative  of  order  5k  will  furnish  a  primitive  group,  since 
the  linear  group  generated  by  K  and  T  is  primitive  (§  101). 
There  are  in  all  9  groups  of  order  5fc  : 


Group       Order  Generating  Substitutions 

13°       5-16*     (ww^BMwO  ;  K,  T; 

14°     10-  16*  "  (tiWeXwifc);  K,  T,  R2; 

15°     20-164>  "  (wwtwew,);  K,  T,  R; 

16°    60-16<£  "  (t*W4)(wi>«);  K,  T,  SB; 

17°           "  "  (uwOCuwe);  K,  T,  BR; 

18°  120-164>  "  (w6wt);  K,  T,  A; 

19°           "  "  (wtwi)  (w3w,) 


20°  360-16^  "  (WiwMwiW*);  K,  T,  AB; 

21°  720-16$  "  (wiw2)',  K,  T,S. 

Here  A=p(l,  i,  i,  1),  B  =  p(l,  1,  1,  —1),  where  p 

(1  -\-i)/V%}   and  R  is  the  transformation  :  x{=  ~7^( 
1  1  1 


125.  There  are  no  more  primitive  groups.  For  a  sub 
group  F'i  of  the  symmetric  group  on  six  letters  contains  an 
invariant  subgroup  F'2  of  order  either  2,  4,  3,  or  9,  unless 
the  order  of  F[  is  a  multiple  of  5.  If  F'z  is  of  order  2  or  4, 
the  corresponding  subgroup  F2  of.F  is  of  order  32<£  or  64<£, 
and  the  group  F\  containing  F2  invariantly  belongs  to  the 
category  of  groups  discussed  in  §  121  (cf.  §  123).  If  F'2 
is  of  order  3,  it  is  generated  by  (w3w6W4)  or  -by  (wiW3w<,) 
In  the  first  case,  FI  contains  a  self-conjugate 


LINEAR  GROUPS  IN  FOUR  VARIABLES  173 

intransitive  subgroup,  namely,  one  generated  by  the 
second  powers  of  F2  (the  transformation  corresponding 
to  (wzWsWt)  is  BR'AB  and  has  the  form  of  Ci,  §  14).  In 
the  second  case,  F2  is  generated  by  K  and  the  trans 
formation  Xi  =  x[,  xz  =  £3,  x3  =  x'4,  z4  =  xi,  and  FI  by  F2  and 
the  transformations  B  and  SR2,  corresponding  to  (wiW2) 
(w3w4)  (W$WQ)  and  (^1^2)  (W^WG)  (wtfjuz),  or  is  a  subgroup  of 
this  group,  excepting  the  possibility  where  the  order  of  F\ 
is  divisible  by  9,  treated  below.  But  this  group  is 
monomial.  Finally,  if  Fi  is  of  order  9,  the  corresponding 
subgroup  F2  of  F  must  be  equivalent  to  the  group  1°,  §  121, 
both  having  an  invariant  subgroup  of  order  16<£,  equiva 
lent  to  K,  and  both  being  of  the  same  order  (§  124,  (d)). 
Hence,  F\  must  be  one  of  the  groups  1°-12°. 

In  conclusion,  we  point  out  that  no  new  groups  arise 
by  enlarging  any  of  the  groups  13°-21°.  For  each  of 
these  groups  contains  a  single  invariant  subgroup  of  order 
16<£,  namely,  K.  This  is  readily  seen  when  we  observe 
that  no  subgroup  of  order  2a  of  the  symmetric  group  on 
the  letters  wi,  .  .  ,  w6  can  be  transformed  into  itself  by 
T=(w2w^WQWSw5).  Accordingly,  a  group  which  contains 
one  of  the  groups  13°-21°  self -con  jugately  must  therefore 
also  contain  K  self-conjugately,  and  is  again  one  of  the 
groups  13°-21°. 


CHAPTER  VIII 

ON  THE  HISTORY  AND  APPLICATIONS  OF 
LINEAR  GROUPS* 

126.  The  theory  of  linear  groups  of  finite  order  may  be 
said  to  have  been  originated  by  F.  Klein,  who  in  1876 
constructed  the  binary  linear  groups  in  order  to  solve  a 
certain  problem  in  the  Theory  of  Invariants.!  Sub 
sequently  he  extended  the  Galois  Theory  of  Algebraic 
Equations  by  the  introduction  of  linear  groups  t  (cf. 
§  127). 

An  important  problem  connected  with  Linear  Differ 
ential  Equations,  namely  the  determination  of  those  equa 
tions  of  this  type  whose  coefficients  are  rational  functions 
of  the  independent  variable  x  and  whose  solutions  are 
algebraic  functions  of  x,  was  in  the  meantime  attacked 
by  various  men,  in  particular  H.  A.  Schwarz,§  L.  FuchsJ 
and  C.  Jordan.  1f  Their  solutions  hinged  upon  the  dis 
covery  of  the  invariants  of  certain  corresponding  linear 
groups  or  the  groups  themselves,  although  the  group- 
notion  was  not  at  first  introduced,  except  by  Jordan. 
This  author  made  the  problem  entirely  one  of  linear 

*  In  the  matter  of  references,  the  following  abbreviations  are  used  in 
the  present  chapter:  Annalen,  2,  9,  12,  28,  50,  52,  60,  71,  for  Mathematische 
Annalen,  Bd.  2  (1870),  9  (1876),  12  (1877,)  28  (1887),  50  (1898),  52  (1899), 
60  (1905),  71  (1912),  respectively;  Crelle,  75,81,  84,  85,  for  Journal  fur  die 
reine  und  angewandte  Mathematik,  Bd.  75  (1873),  81  (1876),  84  (1878),  85 
(1878),  respectively;  Transactions,  VI,  VIII,  XIV,  for  Transactions  of  the 
American  Mathematical  Society,  VI  (1905),  VIII  (1907),  XIV  (1913); 
Icosaeder,  for  Vorlesungen  ilber  das  Icosaeder,  by  F.  Klein,  Leipzig,  1884; 
Valentiner,  for  De  endelige  Transformations-Cruppers  Theori,  Videnskabernes 
Selskabs  Skrifter,  (6),  Copenhagen,  1889. 

t  Annalen,  9,  pp.  183-208. 

t  Cf.  Icosaeder.  \\  Ibid.,  81,  pp.  97-142;  85,  pp.  1-25. 

§  Crelle,  75,  pp.  292-335.  1  Ibid.,  84,  pp.  89-215. 

174 


APPLICATIONS  OF  LINEAR  GROUPS  175 


groups;  and,  having  proved  a  proposition  concerning  the 
order  of  such  groups  (§  74),  he  was  able  to  state  a  general 
theorem  about  the  degree  of  the  algebraic  equation  whose 
roots  satisfy  a  linear  differential  equation  (cf.  §  128).* 

Having  briefly  mentioned  the  important  applications 
of  linear  groups,  outside  of  general  group-theory  proper, 
we  shall  give  a  survey  of  the  .progress  of  the  theory  up 
to  the  present.  As  already  stated,  Klein  began  by  the 
construction  of  the  groups  in  two  variables.  This  was 
followed  by  different  determinations  of  the  same  groups 
by  P.  Gordan,  Jordan,  and  H.  Valentiner,f  and  of  the 
groups  in  three  variables  by  Jordan  {  and  Valentiner§. 
A  general  theory  for  any  number.  of  variables  was  out 
lined  by  the  former  and  applied  by  him  with  partial  suc 
cess  to  the  groups  in  four  variables.il  In  addition, 
special  groups  of  the  latter  category  that  arose  in  analysis, 
or  special  classes  of  such  groups,  were  discussed  by  other 
mathematicians  (Klein,  ^f  Goursat,**  Bagnera,ff  etc.). 
The  complete  determination  of  the  groups  in  four  variables 
(aside  from  intransitive  and  monomial  types)  was  carried 
through  by  the  author.Jt  More  recently,  H.  H.  Mitchell 
has  given  a  partial  determination  of  these  groups  by  means 
of  a  classification  based  upon  certain  geometrical  prop 
erties^ 

There  are,  in  the  main,  four  distinct  principles  em 
ployed  in  the  determination  of  the  groups  in  2,  3,  or  4 

*  Ibid.,  p.  91.  t  Crelle,  84,  pp.  125-215. 

t  See  footnote  to  §  52.  §  Valentiner  (ct.  footnote  above). 

||  Atti  della  Reale  Accademia  della  Scienze  fisiche  e  matematiche  di 
Napoli,  t.  8  (1879). 

H  Annalen,  2,  pp.  198  ff.;    28,  pp.  504  ff.,  etc. 

**  Annales  scientifiques  del'Ecole  Normale  Supcrieure  (3),  t.  6  (1889), 
pp.  9-102. 

ft  Rendiconti  del  Circolo  Matematico  di  Palermo,  t.  15  (1901),  161- 
309;  t.  19  (1905),  pp.  1-56. 

it  Annalen,  60,  pp.  204-31;    Transactions,  6,  pp.  232-36. 

§§  Transactions,  14,  pp.  123-42. 


176  FINITE  COLLINEATION  GROUPS 

variables:  (a)  the  original  geometrical  process  of  Klein 
(chap,  iii);  (6)  the  processes  leading  to  a  diophantine 
equation,  which  may  be  approached  analytically  (Jordan, 
§59),  or  geometrically  (Valentiner,  tBagnera,  Mitchell); 

(c)  a  process  involving  the  relative  geometrical  properties 
of  transformations  which  represent   "homologies"   and 
like  forms  (Valentiner,  Bagnera,  Mitchell;  cf.  §§  80,  103); 

(d)  a  process  developed  from  the  properties  of  the  multi 
pliers  of  the  transformations,  which  are  roots  of  unity 
(Blichfeldt,  §§  63-68).     A  new  principle  has  been  added 
recently  by  Bieberbach,  though  it  had  already  been  used 
by  Valentiner  in  a  certain  form  (see  footnote  p.  97).* 

Independent  of  these  principles  stands  the  theory 
of  group  characteristics,  of  which  G.  Frobenius  is  the 
discoverer  (chap.  vi).  Important  additions  have  been 
made  by  I.  Schur,  W.  Burnside,  and  T.  Molien.f  Re 
cently  L.  E.  Dickson  has  developed  a  theory  of  group 
characteristics  for  modular  groups,  t 

In  conclusion  we  shall  dwell  for  a  moment  on  an 
important  question  connected  with  linear  groups:  What 
is  the  arithmetical  nature  of  the  elements  involved  in  such 
groups?  The  following  theorem  has  been  established 
(Maschke,§  Burnside, II  Schur  1f):  "The  n  variables  may 
be  so  chosen  that  every  element  in  the  matrices  is  a 
cyclotomic  number  (that  is,  a  linear  function  of  roots  of 
unity  with  coefficients  which  are  rational  numbers). 
Possible  exceptions  to  this  rule  can  occur  only  if  the 

*  The  author  has  amplified  this  principle  and  hopes  shortly  to  publish 
his  results  (cf.  §  74). 

t  See  footnote  to  §  84. 

t  Transactions,  8,  pp.  389-398;  Bulletin  of  the  American  Mathematical 
Society  (2),  XIII  (1907),  477-88. 

§  Annalen,  50,  pp.  492-98. 

||  Proceedings  of  the  London  Mathematical  Society,  (2),  III  (1905),  239. 

^  Sitzungsberichte  der  Konigl.-Preuss.  Akademie  der  Wissenschaften, 
190G,  pp.  164  fl. 


APPLICATIONS  OF  LINEAR  GROUPS  177 

characteristic  equation  (-0)n+A(-0)n-l+  .  .  .  =  0 
(§  23)  of  every  transformation  of  the  group  has  the  form 
of  a  perfect  kth  power,  k  being  greater  than  unity  and  a 
factor  of  n."  In  addition,  Schur  has  proved  that  the  ele 
ments  are  algebraic  integers  (§  134),  when  the  variables  are 
suitably  chosen.*  In  the  case  of  a  monomial  group,  W.  A. 
Manning  has  demonstrated  that  under  a  proper  choice  of 
variables,  every  non-zero  element  is  a  root  of  unity,  f 

127.  We  shall  now  outline  briefly  the  main  points  in 
the  Galois  theory  of  equations  and  Klein's  extension 
thereof.t  Let  A  =  xn-ciXn-1-\-C2Xn-2-  .  .  =0  be  an 
equation  whose  coefficients  ci,  .  .  ,  cn  are  numbers  in  a 
domain  R.  (A  domain  R  consists  of  all  numbers  which 
are  rational  functions  with  rational  coefficients  of  certain 
specified  numbers  ki,  .  .  ,  km  defining  R;  for  instance, 
the  domain  defined  by  a  single  rational  number  k,  different 
from  zero,  is  the  aggregate  of  all  rational  numbers,  includ 
ing  zero.)  The  equation  A  =  0  is  irreducible  in  R  if  A  is 
irreducible;  that  is,  if  no  factor  of  A  exists  which  is  a 
polynomial  in  x  of  degree  <n  and  with  coefficients  in  R. 
Under  this  condition,  integers  mi,  .  .  ,  mn  can  be  found 
such  that  the  n\  expressions  y\t  .  .  yn\,  obtained  from 
yi  =  niiXi+  .  .  +mnxn  by  subjecting  x\,  .  .  ,  xn  to  all 
the  n\  possible  permutations,  are  all  distinct.  Consider 
the  function  B=(y—yi)(y  —  yz)  .  .  .  (y  —  yn\).  The  co 
efficients  of  the  different  powers  of  y  are  all  symmetric 
functions  of  xi,  .  .  ,  xn  and  are  therefore  numbers  in  R. 
Let  B'=(y-yi)(y-y2)  .  .  .  (y  —  yg)  be  a  factor  of  B, 
irreducible  in  R;  the  equation  B'  =  0  is  called  a  Galoisian 
resolvent  of  A  =  0  for  the  domain  R.  Its  roots  y\,  .  .  ,  ya 
are  obtained  from  one  of  them  by  subjecting  x\,  .  .  ,  xn 

*  Annalen,  71,  p.  365. 

t  Bulletin  of  the  American  Mathematical  Society  (2),  XII  (1905), 
77-79. 

I  Of.  Miller,  Blichfeldt,  and  Dickson,  Theory  and  Applications  of 
Finite  Groups,  New  York,  1916,  pp.  279  fl. 


178  FINITE  COLLINEATION  GROUPS 

to  g  substitutions  forming  a  substitution  group  G,  called 
the  group  of  the  equation  A  —  Q  for  the  domain  R.  Then, 
according  to  the  Galois  theory,  if  a  rational  function  of 
the  roots  xi,  .  .  ,  xn  with  coefficients  in  R  (or  in  an 
enlarged  domain  R'  containing  R)  is  an  absolute  invariant 
under  G,  this  function  equals  a  number  in  R  (or  R'). 
Again,  if  G  has  an  invariant  subgroup  of  index  p  (a  prime 
number),  the  resolvent  B'  =  Q  can  be  broken  up  into  factors 
by  the  "  ad  junction"  to  our  domain  R  of  the  roots  of 
unity  of  index  p  (i.e.,  including  these  roots  among  the 
defining  numbers  of  the  domain),  as  well  as  a  radical 
l/K,  where  K  is  a  number  in  the  enlarged  domain  R. 

In  Klein's  theory,  we  construct  the  regular  substitu 
tion  group  H  of  order  g  consisting  of  the  permutations 
that  take  place  among  the  roots  of  E'  =  0  when  x\,  .  .  ,  xn 
are  subjected  to  the  substitutions  of  G.  This  group  H  is 
intransitive  as  a  linear  group  (§  96),  breaking  up  into 
component  groups  H',  H",  .  .  ,  of  respectively  n't 
n",  .  .  variables,  linear  functions  of  y\,  .  .  ,  yg.  At 
least  one  of  the  numbers  n',  n" ,  ,  .  is  unity  (say  nf  =  1), 
the  corresponding  variable  being  2/1+2/2+  •  •  +2/0-  If 
H  contains  an  invariant  subgroup  of  index  p  (a  prime 
number),  then  an  additional  number  n",  .  .  is  unity, 
say  n"=l.  The  corresponding  variable  is  of  the  form 
Y=y'+0y"+  .  .  +  Qp-iy  (P-D,  where  0  is  a  root  of 
unity  of  index  p,  and  y',  y",  .  .  are  each  the  sum  of  g/p 
of  the  letters  y\,  .  .  ,  yg.  The  transformations  of  H" 
are  here  of  the  form  Y  =  Oj  Y',  so  that  Yp  —  K  is  an  absolute 
invariant  of  H"  and  therefore  also  of  H  and  G.  In  agree 
ment  with  the  Galois  theory,  K  is  a  number  in  the  domain 
R'  obtained  by  adjoining  0  to  R,  and  we  have  Y  =  fy K. 

On  the  other  hand,  if  H  is  simple,  then  none  of  the 
numbers  n",  .  .  equals  unity.  Consider  in  this  case  the 
group  H"  in  n"  variables.  Its  invariants  are  invariants 
of  H  and  G,  and  are,  accordingly,  equal  to  numbers 


APPLICATIONS  OF  LINEAR  GROUPS  179 

in  the  domain  R"  obtained  by  adjoining  to  R  those 
coefficients  of  the  various  products  of  x\,  .  .  ,  xn  in 
volved  in  the  invariants  in  question  which  are  not  already 
numbers  of  R  (these  coefficients  depend  only  on  the  in 
tegers  mi,  .  .  ,  mn  and  the  " multiplication  table"  of 
the  group  H  and  are  presumably  "cyclotomic"  numbers; 
cf.  §§  95,  126).  If  therefore  we  are  able  to  evaluate,  in 
some  way,  the  n"  variables  of  H"  from  these  invariants, 
we  shall  have  obtained  that  many  new  unsymmetrical 
functions  of  the  roots  of  A  —  0,  and  our  problem  has  been 
reduced  correspondingly.  This  evaluation  is  what  is 
called  the  binary,  ternary,  etc.,  form-problem,  in  the  cases 
n"  =  2,  3,  etc.  The  form-problem  of  the  first  order  (n"  =  1) 
is  solved  by  the  extraction  of  a  radical  (y'K),  and  an 
equation  can  be  "solved  by  radicals"  if  the  form-problems 
of  the  successive  resolvents  are  all  of  the  first  order. 
Consider  the  "general"  quintic  equation 

.    .    =0. 


If  the  numbers  defining  the  domain  R  be  the  coefficients 
d,  .  .  ,  c5  and  the  square  root  of  the  discriminant  of 
A  =  0,  the  group  of  this  equation  becomes  the  alter 
nating  group  on  5  letters.  The  numbers  ft",  .  .  , 
corresponding  to  the  different  non-equivalent  groups 
H",  .  .  are  3,  3,  4,  5  (cf.  Exercise  1,  §  97).  The  general 
quintic  can  accordingly  be  reduced  by  means  of  a  ternary 
form-problem.  But,  if  the  equation  A  =  0  is  first  thrown 
into  the  form  A'=v>-\-av2-{-bv-{-c  =  Q  by  means  of  the 
so-called  Tschirnhausen  transformation  (requiring  the 
extraction  of  a  square  root  in  addition  to  rational  opera 
tions  performed  on  the  coefficients  ci,  .  .  ,  c5),  its  reduc 
tion  is  made  to  depend  upon  a  binary  form-problem. 
For,  the  60  functions  obtained  by  permuting  the  roots 
v\,  vz,  .  .  ,  fs  of  A'  =  0,  according  to  the  alternating 
group  on  5  letters,  in  the  function 


180  FINITE  COLLINEATION  GROUPS 

where  6  is  a  fifth  root  of  unity,  are  all  linear  fractional 
functions  of  z\  (when  account  is  taken  of  the  relations 


namely  the  functions  representing  the  60  transformations 
of  the  linear  fractional  group  (§11)  corresponding  to  the 
binary  icosahedral  group  (E),  §  58  (second  form).  The 
invariants  of  this  group  are  accordingly  equal  to  numbers 
in  the  domain  defined  by  0,  the  coefficients  of  A'  =  Q,  and 
the  square  root  of  the  discriminant  of  A'  =  Q. 

Extending  these  results,  Klein  has  made  the  reduction 
of  the  general  equations  of  degrees  6  and  7  depend  upon 
a  ternary  and  quaternary  form-problem  respectively,  the 
corresponding  linear  groups  being  the  Valentiner  group 
of  order  360<£*  and  a  group  of  order  7!<£/2  in  four  variables 
first  constructed  by  Klein,  f  Beyond  that,  the  general 
equation  of  degree  n^S  can  be  reduced  by  a  form-problem 
of  order  n—  1,  but  not  lower  (Wimanf). 

128.  The  application  of  linear  groups  to  linear  differ 
ential  equations  having  algebraic  solutions  will  now  be 
briefly  explained.  By  the  "  domain  R"  should  here  be 
understood  the  aggregate  of  all  constants  (real  or  complex, 
algebraic  or  transcendental)  and  all  rational  functions  of 
a  single  complex  variable  x,  and  by  an  "  algebraic  func 
tion  of  x"  shall  be  meant  a  function  which  is  a  root  of  an 
algebraic  equation  whose  coefficients  are  functions  in  R. 

(11J 

Consider  the  differential  equation  (y'  =  -r-,  etc.): 
(1)  y"+py'+gy=0, 

where  p  and  q  are  functions  in  R.  Let  it  be  given  that 
the  equation  is  "  irreducible"  in  R  (i.e.,  no  solution  of 

*  Valentiner  (cf.  footnote  at  the  opening  of  §  126),  p.  198;  the  group 
is  (I),  §  82. 

t  Annalen,  28,  p.  519.  J  Ibid.,  52,  p.  243. 


APPLICATIONS  OF  LINEAR  GROUPS  181 

(1)  satisfies  an  equation  y'+ry  =  Q,  where  r  is  in  R),  and 
also  that  two  independent  solutions  y\,  yz  of  (1)  are  alge 
braic  functions  of  x.  Then  if  s  is  an  arbitrary  constant 
(not  specified),  the  function 


is  a  solution  of  the  differential  equation 
(2)  vf+v*+pv+q  =  0, 

and  is  at  the  same  time  a  root  of  an  algebraic  equation 
(3) 


with  coefficients  Ai,  .  .  in  R.  If  we  assume  that  (3)  is 
irreducible  in  R,  it  follows  that  its  roots  are  all  solutions  of 
(2).  Hence,  any  such  root  must  be  of  the  form 


where  s'  is  a  constant.  On  the  other  hand,  the  roots  of 
(3)  are  all  of  the  form  v\  as  regards  s;  it  is  therefore  easy 
to  prove  that  s=(as'+  6)/(cs'+d),  where  a,  6,  c,  d  are 
certain  (known)  constants.  But  this  is  the  typical  form 
of  a  linear  fractional  transformation  T2,  and  we  may 
write  (vi)Tz  =  vz. 

The  roots  of  (3)  are  accordingly  obtained  from  v\ 
by  subjecting  s  to  m  linear  fractional  transformations 
Ti,  .  .  ,  Tm.  It  is  readily  seen  that  these  transformations 
form  a  group  G.  For,  writing  s'  originally  instead  of  s 
in  vi,  the  roots  of  (3)  would  evidently  again  be  derived 
from  the  new  form  of  v\  (which  is  the  original  #2)  by 
subjecting  s'  to  the  same  linear  fractional  transforma 
tions  Ti,  .  .  ,  Tm.  Accordingly,  the  products  T2Ti, 
T\,  .  .  y  T2Tm  are  Ti,  .  .  ,  Tm  over  again  in  some 
order. 


182  FINITE  COLLINEATION  GROUPS 

It  follows  that  m  is  one  of  the  numbers  g,  2g,  12,  24, 
60  (cf.  §§11,  56-58).  However,  s  can  be  specialized  so  as 
to  reduce  this  number  m.  For,  let  G  contain  a  subgroup 
H  of  order  h,  which  possesses  a  linear  invariant  when  it 
is  written  as  a  linear  group  in  two  " variables'7  s,  t;  by 
a  suitable  choice  of  these  variables  we  can  cause  this 
invariant  to  be  t.  If  then  we  put  s  =  0  in  (3),  the  resulting 
equation  will  have  h  roots  all  equal  to  2/2/2/2,  and  (3)  will 
break  up  into  h  equal  factors.  Omitting  the  case-m/A  =  1, 
we  finally  find  that  the  degree  of  the  equation  (3)  may  be 
put  equal  to  2,  4,  6,  12,  in  the  cases  where  the  "mono- 
dromie"  groups  are  the  linear  fractional  groups  corre 
sponding  to  (B),  (C),  (D),  (E),  §§  56-58,  respectively. 

Similar  results  are  obtained  for  linear  differential 
equations  of  any  order.  In  the  case  of  an  irreducible 
linear  homogeneous  differential  equation  of  the  third 
order,  whose  coefficients  are  in  R  and  whose  solutions  are 
algebraic,  the  degree  of  the  algebraic  equation  with  coeffi 
cients  in  R  satisfied  by  the  function  y'/y  of  a  certain 
solution  y  of  the  differential  equation,  is  3,  6,  6,  9,  9,  6, 
36,  21,  according  as  the  corresponding  linear  fractional 
group  is  isomorphic  with  the  linear  group  (C),  .  .  ,  or 
(J),  §§  76,  79,  82.  This  is  Jordan's  theorem  (cf.  §  126), 
as  modified  by  adopting  Painleve*'s  substitution  (v  =  y'/y).* 

The  complete  determination  of  the  types  of  equations 
(1)  whose  monodromie  group  is  the  icosahedral  group  (E), 
§58,  has  been  carried  through  by  Klein,  chiefly  by  aid 
of  the  "Schwarzian  derivative" :  y'"/y'-  l(y"/y'Y,  which 
remains  unaltered  when  y  is  subjected  to  a  linear 
fractional  transformation  with  constant  coefficients,  f 
Similar  methods  have  been  applied  to  the  equations  of  the 
third  order  by  Painleve**  and  Boulanger.J 

*  Comptes  rendua,  Paris,  104  (1887),  pp.  1829-32;  ibid.,  105  (1888), 
pp.  58-61. 

t  Annalen,  12,  pp.  167-80. 

I  Journal  de  I' E  cole  Polytechnique  (2),  4  (1898),  pp.  1-122. 


APPENDIX 

129.  Congruences.    The  symbol 
(1)  A  =  B        (mod  A;) 

is  read  "A  is  congruent  to  B  modulo  fc"  and  takes  the 
place  of  the  equation 

A=B+Ck 

where  C  is  a  positive  or  negative  integer  or  zero.  The 
congruence  is  used  in  preference  to  the  equation  whenever 
it  is  immaterial  just  what  the  value  of  C  is.  The  following 
rules  apply:  if  A =B  and  C=D  (mod  k),  then 

A-B=Q          (mod/c), 

mA=mB       (mod  k)  when  m  is  an  integer, 
A±C=B±D  (mod/c), 

AC=BD      (mod  /c). 

The  least  absolute  remainder  (positive  or  negative) 
obtained  when  a  number  A  is  divided  by  k  is  called  the 
remainder  of  A  (mod  k) . 

Remark. — In  chap,  iv,  the  congruence  notation  (1)  is 
extended  to  the  case  where  A  and  B  are  sums  of  roots  of 
unity,  the  meaning  now  being  that  "A  equals  £±A;X(the 
sum  of  a  finite  number  of  roots  of  unity)/' 

130.  Roots  of  a  congruence.    Let  a,  6,  c  be  given 
integers  and  p  a  prime  number,  then  by  "a  root  of  the 
congruence  " 

axz+bx-\-c=Q        (mod  p) 

is  meant  such  an  integer  (if  any)  which,  when  substituted 
for  x,  will  cause  the  left-hand  member  to  become  an 
integral  multiple  (positive  or  negative)  of  p,  or  zero. 

183 


184 


FINITE  COLLINEATION  GROUPS 


For  example,  the  congruence  x2— 1  =  0  (mod  2)  has 
for  root  every  odd  number.  These  are  all  =1  (mod  2); 
we  therefore  say  that  the  congruence  given  has  just  one 
root  (mod  2).  If  p>2,  the  congruence  x2— 1  =  0  (mod  p) 
has  two  roots  (namely  ±1).  The  congruence  z2=2 
(mod  3)  has  none. 

We  have  the  following  theorem: 

A  congruence  of  the  nih  degree  (axn+  ...  =0)  has 
at  most  n  roots  (mod  p),  p  being  a  prime  number. 

131.  Indeterminate  equations   of   the   first    degree. 
The  congruence 

ax=c         (mod  b) 

is  equivalent  to  the  indeterminate  equation 
ax+by  =  c, 

to  be  satisfied  by  integral  values  of  x  and  y. 

If  the  highest  common  factor  of  a  and  b  divides  c, 
there  is  always  a  solution  of  this  equation.  In  particu 
lar,  if  c=l,  there  is  a  solution  if  a  and  b  are  prime  to 
each  other. 

132.  On  a  certain  class  of  determinants.    The  deter 
minant 


f(xi,   .    .    .   ,  xn)  = 


has  the  value 

f=(Xi-Xi)'(Xt-Xi)     .     .      -     (Xn- 
(Xn-Xz)    «...     (X 


1 

1 

...    1 

Xi 

X2 

...      xn 

xl 

xl 

...     xl 

z?-1 

xl~l 

...  zr1 

APPENDIX 

and  is  a  factor  of  the  determinant 


185 


I'///  .« ///  >• /•/ 

The  value  of  the  quotient  F/f,  when  x\,  .  .  .  ,  xn  all 
have  been  replaced  by  1,  may  be  found  by  treating  the 
fraction  F/f  as  an  "indeterminate  form  0/0."  We  put 
xi  =  I  and  let  Xz  approach  1  as  a  limit ;  the  value  of  F/f  will 
then  become 


Next  we  let  x3  approach  1,  etc.     The  final  result  is  the 
ratio  of  the  two  following  determinants: 


1 

a 

a(a— 

D 

o(o-l)(a-2)    ... 

1 

b 

6(6- 

D 

6(6-l)(6-2)    ... 

1 

m 

m(m  — 

1) 

m(m— 

l)(m-2) 

• 

1 

0 

0 

0 

1 

1 

0 

. 

0 

1 

2 

2! 

0 

1 

n-l      (w-l)(w-2) 

.  .  .   (»'- 

1)! 

We  find 


namely  D  :  [21  3!   . 


D  = 


1 
1 

a 
b 

a2       ... 

1 

m 

m2      ... 

=/(a,  6,  . . .  ,  m). 


186  FINITE  COLLINEATION  GROUPS 

133.  On  roots  of  unity.    A  solution  of  the  equation 
*•-!, 

n  being  a  positive  integer,  is  called  a  root  of  unity.  A  solu 
tion  a  is  in  particular  called  a  primitive  nth  root  of  unity, 
if  n  is  the  least  integer  for  which  an  =  1.  In  such  a  case 
n  is  called  the  index  of  the  root. 

The  following  theorems  are  useful : 

1°.  The  product  or  ratio  of  two  roots  of  unity  is  a 
root  of  unity. 

2°.  Any  positive  or  negative  rational  power  of  a 
root  of  unity  is  again  a  root  of  unity. 

3°.  If  n  is  the  index  of  a  root  a,  and  m  a  positive 
integer,  the  index  of  am  is  n/d,  where  d  is  the  highest 
common  factor  of  n  and  m. 

4°.  If  the  index  of  a  root  0  is  n  =  ab,  where  a  and  b  are 
two  integers  which  are  prime  to  each  other,  then  it  is 
possible  to  find 'a  root  of  index  a,  say  a,  and  one  of  index 
6,  say  ft,  such  that  Q  =  afi. 

As  is  customary,  we  write  <o,  <o2  for  the  roots  of  index  3, 
and  i,  —i  for  the  roots  of  index  4. 

5°.  If  a  is  a  root  of  index  n,  then  the  n  solutions  of 
xn=l  are  1,  a,  a2,  .  .  ,  an~1)  and  we  have 


6°.  Theorem  of  Kronecker.  * — For  the  proper  handling 
of  a  certain  class  of  equations  a  very  effective  theorem  due 
to  KrOnecker  is  necessary.  We  shall  not  make  a  formal 
statement  of  the  theorem,  but  explain  its  meaning  by 
implication. 

Trie  class  of  equations  referred  to  are  all  of  the  form 

=  0,   ai,   .    .    .   ,  ak  being  roots  of  unity,  and  the 

*"Memoires  sur  les  facteurs  irroductiblos  de  1'exprossion  xM-l," 
Journal  de  Afathtmatique  pures  et  appliqutea,  I,  t.  19  (1854),  pp.  178  ft. 


APPENDIX  187 

question  involved  is  this  :  If  we  know  nothing  about  these 
roots  except  their  number  k,  what  can  be  inferred 
concerning  their  values?  Kronecker's  theorem  implies 
that  the  k  roots  fall  into  sets,  each  containing  a  prime 
number  of  roots  the  sum  of  which  equals  zero.  Moreover, 
if  p  is  the  number  of  roots  in  any  one  set,  and  if  a  is  a 
root  of  index  p,  then  the  roots  of  the  set  are  e,  ea, 
ca2,  .  .  .  ,  eaP-1,  where  e  is  an  unknown  root  of  unity. 
We  shall  discuss  in  full  the  cases  k  =  3,  4,  5. 

/c  =  3:  ai-f-a2-j-a3  =  0.     Here  we  have  a2  =  a1co,  a3=a1w2. 

k  =  4:  ai-j-a2+a3+a4  =  0.  We  have  two  sets  of  two 
roots  each,  say  «i  4-^2  =  0,  a3-f-a4  =  0. 

ft  =  5:  ai-f-a2+a3-fa4-{-a5  =  0.  There  are  two  possi 
bilities:  one  set  only,  or  two  sets  containing 
3  and  2  roots  respectively.  If  J3  represents  a 
primitive  5th  root,  and  y,  8  roots  of  unknown 
indices,  the  two  cases  are  respectively  given 
by 


(y-y)  + 

By  means  of  Kronecker's  theorem  the  following  can 
be  proved: 

7°.  If  N  represents  the  sum  of  a  finite  number  of  roots 
of  unity  and  k  an  integer,  and  if  it  be  known  that  N/k  is 
an  algebraic  integer  (§  134),  then  N/k  equals  the  sum  of  a 
finite  number  of  roots  of  unity. 

More  definitely,  the  roots  in  N  can  be  arranged  in  two 
sets  such  that  the  sum  of  those  in  one  set  vanishes  and 
those  in  the  other  set  are  each  repeated  k  (or  a  multiple 
of  k)  times. 

8°.  By  Demoivre's  Theorem: 

(cos  0+i  sin  0)n  =  cos  nO+i  sin  nO, 


188  FINITE  COLLINEATION  GROUPS 

the  roots  of  unity  of  index  n  can  be  expressed  in  the  form 

a™  =  cos™  360°+z  sin-  360°, 
n  n 

where  m  represents  in  turn  every  integer  prime  to  and  less 
than  n. 

The  conjugate-imaginary  of  am  is  accordingly  a~m. 

134.  On  algebraic  integers.  An  algebraic  integer  is 
a  number  which  satisfies  an  equation  of  the  form 

xn+aixn~l+   .    .   +an-ix+an  =  0, 

where  a\,  .  .  ,  an  are  positive  or  negative  integers  or 
zero. 

If  a  and  ft  are  algebraic  integers,  and  k  an  ordinary 
integer,  then  a+/3,  a/3,  and  ka  are  algebraic  integers. 

If  a/6  is  an  algebraic  integer,  and  at  the  same  time  a 
and  b  are  ordinary  integers,  then  a/b  must  be  an  ordinary 
integer. 


INDEXES 


GENERAL  INDEX 

(Numbers  refer  to  pages] 


Abelian  groups,  26;  43-45 
Abstract  groups,  30,  n. 
Algebraic  integer,  188 
Alternating  groups,  54;  60-61 
Associative  law,  5;  30 

Binary  groups,  63-75 

Canonical  form,  3;  24-27 
Change  of  variables,  15-17 
Characteristic  equation,  27-28 
Characteristics,    28;     117;     of 
inverse  and  conjugate  trans 
formations,    and    of    substi 
tutions,  118;    general  theory, 
116-38 

Class  of  a  substitution  group  is 
the  least  number  of  letters  that 
are  replaced  by  different  letters 
by  a  substitution  of  the  group 
(except  the  identity) 
Collineations  and  collineation 

groups,  10-12 
Commutative  law,  5;   31 
Components  of  an  intransitive 

linear  group,  117 
Composition  of  two  groups,  125 
Congruences,  183-84 
Conjugate-imaginary  groups,  18 
Conjugate  operators,  sets,  and 

subgroups,  36-38 
Cycle  of  a  substitution,  52 
Cyclotomic  number,  179 

Degree  of  a  substitution  group 
is  the  number  of  distinct  letters 
used  in  the  substitutions  of  the 
group 


Determinant  of  a  linear  trans 
formation,  2;  13,  exs.  3,  4 

Differential  equations  having 
algebraic  solutions,  180-82 

Dihedral  group,  70 

Diophantine  equation,  75 

Domain,  177,  180 

Equation    of    the    fifth    degree, 

179-80 

Equivalent  groups,  64;  129;  135 
Even  substitutions,  53 

Factor  groups,  42-43 
Finite  groups,  33 
Form  problem,  179 

Galois'  theory  of  equations,  with 
Klein's  extension,  177-80 

Galoisian  resolvent,  177 

Generators,  9-10;  33;  39,  ex.  3; 
139,  2° 

Group  characteristics,  116-38 

Group-matrix,  133-35 

Group  of  an  equation,  178 

Group  of  similarity-transforma 
tions,  13,  ex.  1 

Groups:  of  linear  transforma 
tions,  7-15;  of  operators,  33; 
117,  2°;  of  substitutions,  54; 
56-59;  of  order  vft,  45-50;  80; 
81;  of  order  p*(f,  137;  of  the 
regular  polyhedra,  69-73 ;  leav 
ing  invariant  a  quadric  sur 
face,  169,  exs.  1,  2 

Hermitian  form,  19 
Hermitian  invariant,  20-21 
Hessian  group,  109 


191 


192 


FINITE  COLLINEATION  GROUPS 


Icosahedral  group,  73 
Identity,  the,  3;  30;  51;  9;  33 
Imprimitive  linear  groups,  76-79 
Imprimitive  substitution  groups. 

55 

Index  of  a  subgroup,  34 
Intransitive  linear  groups,  17 
Intransitive  substitution  groups, 

55 

Invariants,  120;  125 
Invariant    operators    and    sub 
groups,  39-40 

Inverse:  of  a  linear  transforma 
tion,  5;  7,  exs.  4,  5,  6;  9;  22; 
of  an  operator,  31;  32,  ex.  5; 
33;  of  a  substitution,  51 
Irreducible  algebraic  equations, 

177 

Irreducible  differential  equa 
tions,  180 

Irreducible  groups,  22-24 
Isomorphism,  40-43;  117,  2° 

Linear  fractional  groups,  13 

Linear  groups,  8 

Linear  transformations,  1-7 

Matrices  of  the  transformations 
of  a  transitive  linear  group, 
135,  exs.  1,  2;  176;  177;  sum 
and  product  of,  4;  119 

Matrix  of  a  linear  transforma 
tion,  2 

Monomial  groups,  77;  80 

Monodromie  group,  182 

Multiplication-table  of  a  group, 
40 

Multipliers  of  a  linear  trans 
formation,  3;  7,  ex.  8;  102, 
exs.  1,  2 

Non-equivalent  groups,  64;    135 

Octahedral  group,  72 
Odd  substitutions,  53 
Operators,  1;  30 


Order:  of  a  linear  group,  8;  82; 
127;  129,  ex.  2;  of  a  linear 
transformation,  6;  of  an 
operator,  32;  35;  of  a  group 
of  operators,  33;  of  a  sub 
group,  34;  of  a  primitive 
linear  group,  89;  92;  103 

Permutations,  50 

Power:  of  a  linear  transforma 
tion,  5;  of  an  operator,  32 

Primitive  linear  groups,  77;  94; 
96;  101;  103 

Primitive  substitution  groups, 
55 

Product:  of  linear  transforma 
tions,  3-5;  of  matrices,  119; 
of  operators,  30;  of  substi 
tutions,  51 

Quaternary  groups,  139-73 
Quotient  groups,  42-43 

Reduced  set,  23 
Reducible  groups,  22-24 
Regular  substitution  group,  59; 

131;  135 
Roots  of  unity^  186 

Schwarzian  derivative,  182 

Self-conjugate  operators  and 
subgroups,  39-40 

Set:  of  generators,  33;  of  non- 
equivalent  component  groups, 
131 

Sets:  of  conjugate  operators,  36; 
of  conjugate  subgroups,  38; 
of  imprimitivity  (of  a  linear 
group),  77;  of  intransitivity 
(of  a  linear  group),  18;  of 
intransitivity  (of  a  substitu 
tion  group),  55 

Similarity-transformations,  3;  7, 
ex.  2;  13,  ex.  1;  18,  ex.  1; 
40,  ex.  5 

Simple  groups,  39;  58;  60-61; 
137;  138,  ex.;  147 


INDEXES 


193 


Subgroups,  8;    34-35;    46;    49, 

ex.  1 
Substitutions,    50;     written    as 

linear  transformations,  1 ;   118 
Sum  of  matrices,  119 
Sylow's     theorem     and     Sylow 

subgroups,  46-50;   80;   147 
Symmetric  group,  54 
Systems  of  imprimitivity  (of  a 

substitution  group),  55.     See 

also  Sets 

Ternary  groups,  104-15 

Tetrahedral  group,  71 

Transform  of  an  operator  is  the 
operator  into  which  the  given 
operator  is  transformed,  36 


Transformations.      See    Linear 

Transformations 
Transitive  linear  groups,  17;  131 
Transitive  substitution  groups, 

55 

Transposition,  53 
Types  of  groups,  140 

Unit  circle,  94 

Unitary  form,  21-22;   24;   27 

Valentiner  group  is  the  group  (I), 
113 

Variety  of  a  linear  transforma 
tion  and  of  an  abelian  group, 
90 


INDEX  OF  AUTHORS 

(Numbers  refer  to  pages] 

Bagnera,  G.,  147,  175,  176  Klein,  F.,  4,  65,  142,  170,  174, 
Bieberbach,  L.,  97,  103,.  176  175,  176,  178,  180,  182 

Blichfeldt,  H.  F.,  29,  80,   102,  Kronecker,  L.,  124,  186,  187 

103,  115,  116,  147,  175,  176 

Boulanger,  A.,  182  LinS>  G-  H-»  60 

Burnside,  W.,  4,  29,  60,  80,  113,  Loewy,  A.,  21 

116,  123,  135,  137,  138,  176  . 

Manning,  W.  A.,  177 

Cole,  F.  N.,  29,  60  Maschke,  H.,  112,  135,  141,  142, 

159,  163,  170,  176 

Demoivre,  A.,  187  Miller,  G.  A.,  29,  60,  113 

Dickson,  L.  E.,  29,  30,  61,  116,      Mitchell,  H.  H.,  115,  175,  176 


176,  177 

Frobenius,  G.,  4,  97,  102,  103, 

116,  119,  124,  176 
Fuchs,  L.,  21,  65,  174 

Galois,  E.,  177 
Gordan,  P.,  65,  175 
Goursat,  E.,  165,  175 

Hermite,  C.,  19 
Hilton,  H.,  29 
Holder,  O.,  60 
Huntington,  E.  V.,  30 


Molien,  T.,  1J6,  176 

5.  H.,  21,  61,  141, 


Netto,  E.,  29 

Painleve,  P.,  182 
Picard,  E.,  21 

Schur,  I.,  4,  103,  116,  119,  129, 

176,  177 

Schwarz,  H.,  174 
Sylow,  L.,  46 

Valentiner,  H.,  21,  65,  73,  97, 
115,  175,  176,  180 


Jordan,  C.,  4,  60,  64,  65,  73,  103,      w 

109,  115,  142,  174,  175,  176,      Weber,  H.,  4 


182 


Wiman,  A.,  180 


194 


61 


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